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Lesson 20: Derivatives and the shapes of curves

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Matthew Leingang
Matthew Leingang

- The document is a section from a calculus course at NYU that discusses using derivatives to determine the shapes of curves. - It covers using the first derivative to determine if a function is increasing or decreasing over an interval using the Increasing/Decreasing Test. It also discusses using the second derivative to determine if a function is concave up or down over an interval using the Second Derivative Test. - Examples are provided to demonstrate finding intervals of monotonicity for functions and classifying critical points as local maxima, minima or neither using the First Derivative Test.

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Section 4.2 Derivatives and the Shapes of Curves V63.0121.002.2010Su, Calculus I New York University June 9, 2010 Announcements No office hours tonight Quiz 3 on Thursday (3.3, 3.4, 3.5, 3.7) Assignment 4 due Tuesday . . . . . .
Announcements No office hours tonight Quiz 3 on Thursday (3.3, 3.4, 3.5, 3.7) Assignment 4 due Tuesday . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 2 / 30
Objectives Use the derivative of a function to determine the intervals along which the function is increasing or decreasing (The Increasing/Decreasing Test) Use the First Derivative Test to classify critical points of a function as local maxima, local minima, or neither. Use the second derivative of a function to determine the intervals along which the graph of the function is . . . . . . concave up or concave 4.2 The Shapes of Curves V63.0121.002.2010Su, Calculus I (NYU) Section June 9, 2010 3 / 30
Outline Recall: The Mean Value Theorem Monotonicity The Increasing/Decreasing Test Finding intervals of monotonicity The First Derivative Test Concavity Definitions Testing for Concavity The Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 4 / 30
Recall: The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that . f(b) − f(a) b . = f′ (c). b−a . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 5 / 30
Recall: The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that . f(b) − f(a) b . = f′ (c). b−a . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 5 / 30
Recall: The Mean Value Theorem Theorem (The Mean Value Theorem) c .. Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that . f(b) − f(a) b . = f′ (c). b−a . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 5 / 30
Why the MVT is the MITC Most Important Theorem In Calculus! Theorem Let f′ = 0 on an interval (a, b). Then f is constant on (a, b). Proof. Pick any points x and y in (a, b) with x < y. Then f is continuous on [x, y] and differentiable on (x, y). By MVT there exists a point z in (x, y) such that f(y) − f(x) = f′ (z) = 0. y−x So f(y) = f(x). Since this is true for all x and y in (a, b), then f is constant. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 6 / 30
Outline Recall: The Mean Value Theorem Monotonicity The Increasing/Decreasing Test Finding intervals of monotonicity The First Derivative Test Concavity Definitions Testing for Concavity The Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 7 / 30
What does it mean for a function to be increasing? Definition A function f is increasing on (a, b) if f(x) < f(y) whenever x and y are two points in (a, b) with x < y. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 8 / 30
What does it mean for a function to be increasing? Definition A function f is increasing on (a, b) if f(x) < f(y) whenever x and y are two points in (a, b) with x < y. An increasing function “preserves order.” Write your own definition (mutatis mutandis) of decreasing, nonincreasing, nondecreasing . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 8 / 30
The Increasing/Decreasing Test Theorem (The Increasing/Decreasing Test) If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b), then f is decreasing on (a, b). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 9 / 30
The Increasing/Decreasing Test Theorem (The Increasing/Decreasing Test) If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b), then f is decreasing on (a, b). Proof. It works the same as the last theorem. Pick two points x and y in (a, b) with x < y. We must show f(x) < f(y). By MVT there exists a point c in (x, y) such that f(y) − f(x) = f′ (c) > 0. y−x So f(y) − f(x) = f′ (c)(y − x) > 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 9 / 30
Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. Solution f′ (x) = 2 is always positive, so f is increasing on (−∞, ∞). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. Solution f′ (x) = 2 is always positive, so f is increasing on (−∞, ∞). Example Describe the monotonicity of f(x) = arctan(x). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. Solution f′ (x) = 2 is always positive, so f is increasing on (−∞, ∞). Example Describe the monotonicity of f(x) = arctan(x). Solution 1 Since f′ (x) = is always positive, f(x) is always increasing. 1 + x2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . So f is decreasing on (−∞, 0) and increasing on (0, ∞). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . So f is decreasing on (−∞, 0) and increasing on (0, ∞). In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 0 .. × .. .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 .. × .. .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
The First Derivative Test Theorem (The First Derivative Test) Let f be continuous on [a, b] and c a critical point of f in (a, b). If f′ (x) > 0 on (a, c) and f′ (x) < 0 on (c, b), then c is a local maximum. If f′ (x) < 0 on (a, c) and f′ (x) > 0 on (c, b), then c is a local minimum. If f′ (x) has the same sign on (a, c) and (c, b), then c is not a local extremum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 13 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . So f is decreasing on (−∞, 0) and increasing on (0, ∞). In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 14 / 30
Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . m . in So f is decreasing on (−∞, 0) and increasing on (0, ∞). In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 14 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 15 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) m . ax . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 15 / 30
Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) m . ax . inm . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 15 / 30
Outline Recall: The Mean Value Theorem Monotonicity The Increasing/Decreasing Test Finding intervals of monotonicity The First Derivative Test Concavity Definitions Testing for Concavity The Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 16 / 30
Concavity Definition The graph of f is called concave up on an interval I if it lies above all its tangents on I. The graph of f is called concave down on I if it lies below all its tangents on I. . . concave up concave down We sometimes say a concave up graph “holds water” and a concave down graph “spills water”. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 17 / 30
Inflection points indicate a change in concavity Definition A point P on a curve y = f(x) is called an inflection point if f is continuous at P and the curve changes from concave upward to concave downward at P (or vice versa). . concave up i .nflection point . . . concave down . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 18 / 30
Theorem (Concavity Test) If f′′ (x) > 0 for all x in an interval I, then the graph of f is concave upward on I. If f′′ (x) < 0 for all x in I, then the graph of f is concave downward on I. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 19 / 30
Theorem (Concavity Test) If f′′ (x) > 0 for all x in an interval I, then the graph of f is concave upward on I. If f′′ (x) < 0 for all x in I, then the graph of f is concave downward on I. Proof. Suppose f′′ (x) > 0 on I. This means f′ is increasing on I. Let a and x be in I. The tangent line through (a, f(a)) is the graph of L(x) = f(a) + f′ (a)(x − a) f(x) − f(a) By MVT, there exists a c between a and x with = f′ (c). So x−a f(x) = f(a) + f′ (c)(x − a) ≥ f(a) + f′ (a)(x − a) = L(x) . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 19 / 30
Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . Solution We have f′ (x) = 3x2 + 2x, so f′′ (x) = 6x + 2. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . Solution We have f′ (x) = 3x2 + 2x, so f′′ (x) = 6x + 2. This is negative when x < −1/3, positive when x > −1/3, and 0 when x = −1/3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . Solution We have f′ (x) = 3x2 + 2x, so f′′ (x) = 6x + 2. This is negative when x < −1/3, positive when x > −1/3, and 0 when x = −1/3 So f is concave down on (−∞, −1/3), concave up on (−1/3, ∞), and has an inflection point at (−1/3, 2/27) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) = x − x = x (5x − 2) 9 9 9 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) =x − x = x (5x − 2) 9 9 9 The second derivative f′′ (x) is not defined at 0 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) =x − x = x (5x − 2) 9 9 9 The second derivative f′′ (x) is not defined at 0 Otherwise, x−4/3 is always positive, so the concavity is determined by the 5x − 2 factor . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) =x − x = x (5x − 2) 9 9 9 The second derivative f′′ (x) is not defined at 0 Otherwise, x−4/3 is always positive, so the concavity is determined by the 5x − 2 factor So f is concave down on (−∞, 0], concave down on [0, 2/5), concave up on (2/5, ∞), and has an inflection point when x = 2/5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
The Second Derivative Test Theorem (The Second Derivative Test) Let f, f′ , and f′′ be continuous on [a, b]. Let c be be a point in (a, b) with f′ (c) = 0. If f′′ (c) < 0, then c is a local maximum. If f′′ (c) > 0, then c is a local minimum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 22 / 30
The Second Derivative Test Theorem (The Second Derivative Test) Let f, f′ , and f′′ be continuous on [a, b]. Let c be be a point in (a, b) with f′ (c) = 0. If f′′ (c) < 0, then c is a local maximum. If f′′ (c) > 0, then c is a local minimum. Remarks If f′′ (c) = 0, the second derivative test is inconclusive (this does not mean c is neither; we just don’t know yet). We look for zeroes of f′ and plug them into f′′ to determine if their f values are local extreme values. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 22 / 30
Proof of the Second Derivative Test Proof. Suppose f′ (c) = 0 and f′′ (c) > 0. Since f′′ is continuous, f′′ (x) > 0 for all x sufficiently close to c. Since f′′ = (f′ )′ , we know f′ is increasing near c. Since f′ (c) = 0 and f′ is increasing, f′ (x) < 0 for x close to c and less than c, and f′ (x) > 0 for x close to c and more than c. This means f′ changes sign from negative to positive at c, which means (by the First Derivative Test) that f has a local minimum at c. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 23 / 30
Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. Remember f′′ (x) = 6x + 2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. Remember f′′ (x) = 6x + 2 Since f′′ (−2/3) = −2 < 0, −2/3 is a local maximum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. Remember f′′ (x) = 6x + 2 Since f′′ (−2/3) = −2 < 0, −2/3 is a local maximum. Since f′′ (0) = 2 > 0, 0 is a local minimum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 10 −4/3 Remember f′′ (x) = x (5x − 2), which is negative when 9 x = −4/5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 10 −4/3 Remember f′′ (x) = x (5x − 2), which is negative when 9 x = −4/5 So x = −4/5 is a local maximum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 10 −4/3 Remember f′′ (x) = x (5x − 2), which is negative when 9 x = −4/5 So x = −4/5 is a local maximum. Notice the Second Derivative Test doesn’t catch the local minimum x = 0 since f is not differentiable there. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
Using the Second Derivative Test II: Graph Graph of f(x) = x2/3 (x + 2): y . . −4/5, 1.03413) ( . . . 2/5, 1.30292) ( . . x . . −2, 0) ( . 0, 0) ( . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 26 / 30
When the second derivative is zero At inflection points c, if f′ is differentiable at c, then f′′ (c) = 0 Is it necessarily true, though? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 27 / 30
When the second derivative is zero At inflection points c, if f′ is differentiable at c, then f′′ (c) = 0 Is it necessarily true, though? Consider these examples: f(x) = x4 g(x) = −x4 h(x) = x3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 27 / 30
When first and second derivative are zero function derivatives graph type f′ (x) = 4x3 , f′ (0) = 0 f(x) = x4 min f′′ (x) = 12x2 , f′′ (0) = 0 . . g′ (x) = −4x3 , g′ (0) = 0 g(x) = −x4 max g′′ (x) = −12x2 , g′′ (0) = 0 h′ (x) = 3x2 , h′ (0) = 0 h(x) = x3 infl. h′′ (x) = 6x, h′′ (0) = 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 28 / 30
When the second derivative is zero At inflection points c, if f′ is differentiable at c, then f′′ (c) = 0 Is it necessarily true, though? Consider these examples: f(x) = x4 g(x) = −x4 h(x) = x3 All of them have critical points at zero with a second derivative of zero. But the first has a local min at 0, the second has a local max at 0, and the third has an inflection point at 0. This is why we say 2DT has nothing to say when f′′ (c) = 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 29 / 30
Summary Concepts: Mean Value Theorem, monotonicity, concavity Facts: derivatives can detect monotonicity and concavity Techniques for drawing curves: the Increasing/Decreasing Test and the Concavity Test Techniques for finding extrema: the First Derivative Test and the Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 30 / 30

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Lesson 20: Derivatives and the shapes of curves

  • 1. Section 4.2 Derivatives and the Shapes of Curves V63.0121.002.2010Su, Calculus I New York University June 9, 2010 Announcements No office hours tonight Quiz 3 on Thursday (3.3, 3.4, 3.5, 3.7) Assignment 4 due Tuesday . . . . . .
  • 2. Announcements No office hours tonight Quiz 3 on Thursday (3.3, 3.4, 3.5, 3.7) Assignment 4 due Tuesday . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 2 / 30
  • 3. Objectives Use the derivative of a function to determine the intervals along which the function is increasing or decreasing (The Increasing/Decreasing Test) Use the First Derivative Test to classify critical points of a function as local maxima, local minima, or neither. Use the second derivative of a function to determine the intervals along which the graph of the function is . . . . . . concave up or concave 4.2 The Shapes of Curves V63.0121.002.2010Su, Calculus I (NYU) Section June 9, 2010 3 / 30
  • 4. Outline Recall: The Mean Value Theorem Monotonicity The Increasing/Decreasing Test Finding intervals of monotonicity The First Derivative Test Concavity Definitions Testing for Concavity The Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 4 / 30
  • 5. Recall: The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that . f(b) − f(a) b . = f′ (c). b−a . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 5 / 30
  • 6. Recall: The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that . f(b) − f(a) b . = f′ (c). b−a . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 5 / 30
  • 7. Recall: The Mean Value Theorem Theorem (The Mean Value Theorem) c .. Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that . f(b) − f(a) b . = f′ (c). b−a . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 5 / 30
  • 8. Why the MVT is the MITC Most Important Theorem In Calculus! Theorem Let f′ = 0 on an interval (a, b). Then f is constant on (a, b). Proof. Pick any points x and y in (a, b) with x < y. Then f is continuous on [x, y] and differentiable on (x, y). By MVT there exists a point z in (x, y) such that f(y) − f(x) = f′ (z) = 0. y−x So f(y) = f(x). Since this is true for all x and y in (a, b), then f is constant. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 6 / 30
  • 9. Outline Recall: The Mean Value Theorem Monotonicity The Increasing/Decreasing Test Finding intervals of monotonicity The First Derivative Test Concavity Definitions Testing for Concavity The Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 7 / 30
  • 10. What does it mean for a function to be increasing? Definition A function f is increasing on (a, b) if f(x) < f(y) whenever x and y are two points in (a, b) with x < y. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 8 / 30
  • 11. What does it mean for a function to be increasing? Definition A function f is increasing on (a, b) if f(x) < f(y) whenever x and y are two points in (a, b) with x < y. An increasing function “preserves order.” Write your own definition (mutatis mutandis) of decreasing, nonincreasing, nondecreasing . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 8 / 30
  • 12. The Increasing/Decreasing Test Theorem (The Increasing/Decreasing Test) If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b), then f is decreasing on (a, b). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 9 / 30
  • 13. The Increasing/Decreasing Test Theorem (The Increasing/Decreasing Test) If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b), then f is decreasing on (a, b). Proof. It works the same as the last theorem. Pick two points x and y in (a, b) with x < y. We must show f(x) < f(y). By MVT there exists a point c in (x, y) such that f(y) − f(x) = f′ (c) > 0. y−x So f(y) − f(x) = f′ (c)(y − x) > 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 9 / 30
  • 14. Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
  • 15. Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. Solution f′ (x) = 2 is always positive, so f is increasing on (−∞, ∞). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
  • 16. Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. Solution f′ (x) = 2 is always positive, so f is increasing on (−∞, ∞). Example Describe the monotonicity of f(x) = arctan(x). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
  • 17. Finding intervals of monotonicity I Example Find the intervals of monotonicity of f(x) = 2x − 5. Solution f′ (x) = 2 is always positive, so f is increasing on (−∞, ∞). Example Describe the monotonicity of f(x) = arctan(x). Solution 1 Since f′ (x) = is always positive, f(x) is always increasing. 1 + x2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 10 / 30
  • 18. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
  • 19. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
  • 20. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
  • 21. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
  • 22. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . So f is decreasing on (−∞, 0) and increasing on (0, ∞). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
  • 23. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . So f is decreasing on (−∞, 0) and increasing on (0, ∞). In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 11 / 30
  • 24. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 25. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 26. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 0 .. × .. .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 27. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 .. × .. .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 28. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 29. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 30. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − . 4/5 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 31. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 32. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 12 / 30
  • 33. The First Derivative Test Theorem (The First Derivative Test) Let f be continuous on [a, b] and c a critical point of f in (a, b). If f′ (x) > 0 on (a, c) and f′ (x) < 0 on (c, b), then c is a local maximum. If f′ (x) < 0 on (a, c) and f′ (x) > 0 on (c, b), then c is a local minimum. If f′ (x) has the same sign on (a, c) and (c, b), then c is not a local extremum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 13 / 30
  • 34. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . So f is decreasing on (−∞, 0) and increasing on (0, ∞). In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 14 / 30
  • 35. Finding intervals of monotonicity II Example Find the intervals of monotonicity of f(x) = x2 − 1. Solution f′ (x) = 2x, which is positive when x > 0 and negative when x is. We can draw a number line: − . 0 .. . + .′ f ↘ . 0 . ↗ . f . m . in So f is decreasing on (−∞, 0) and increasing on (0, ∞). In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 14 / 30
  • 36. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 15 / 30
  • 37. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) m . ax . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 15 / 30
  • 38. Finding intervals of monotonicity III Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). Solution f′ (x) = 2 x−1/3 (x + 2) + x2/3 = 3 x−1/3 (5x + 4) 3 1 The critical points are 0 and and −4/5. − . × .. . + . −1/3 x 0 . − . 0 .. . + 5 .x+4 − . 4/5 .. + 0 − × .. . . . . .. + .′ (x) f ↗ . − ↘ . . 4/5 . 0 ↗ . f .(x) m . ax . inm . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 15 / 30
  • 39. Outline Recall: The Mean Value Theorem Monotonicity The Increasing/Decreasing Test Finding intervals of monotonicity The First Derivative Test Concavity Definitions Testing for Concavity The Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 16 / 30
  • 40. Concavity Definition The graph of f is called concave up on an interval I if it lies above all its tangents on I. The graph of f is called concave down on I if it lies below all its tangents on I. . . concave up concave down We sometimes say a concave up graph “holds water” and a concave down graph “spills water”. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 17 / 30
  • 41. Inflection points indicate a change in concavity Definition A point P on a curve y = f(x) is called an inflection point if f is continuous at P and the curve changes from concave upward to concave downward at P (or vice versa). . concave up i .nflection point . . . concave down . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 18 / 30
  • 42. Theorem (Concavity Test) If f′′ (x) > 0 for all x in an interval I, then the graph of f is concave upward on I. If f′′ (x) < 0 for all x in I, then the graph of f is concave downward on I. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 19 / 30
  • 43. Theorem (Concavity Test) If f′′ (x) > 0 for all x in an interval I, then the graph of f is concave upward on I. If f′′ (x) < 0 for all x in I, then the graph of f is concave downward on I. Proof. Suppose f′′ (x) > 0 on I. This means f′ is increasing on I. Let a and x be in I. The tangent line through (a, f(a)) is the graph of L(x) = f(a) + f′ (a)(x − a) f(x) − f(a) By MVT, there exists a c between a and x with = f′ (c). So x−a f(x) = f(a) + f′ (c)(x − a) ≥ f(a) + f′ (a)(x − a) = L(x) . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 19 / 30
  • 44. Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
  • 45. Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . Solution We have f′ (x) = 3x2 + 2x, so f′′ (x) = 6x + 2. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
  • 46. Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . Solution We have f′ (x) = 3x2 + 2x, so f′′ (x) = 6x + 2. This is negative when x < −1/3, positive when x > −1/3, and 0 when x = −1/3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
  • 47. Finding Intervals of Concavity I Example Find the intervals of concavity for the graph of f(x) = x3 + x2 . Solution We have f′ (x) = 3x2 + 2x, so f′′ (x) = 6x + 2. This is negative when x < −1/3, positive when x > −1/3, and 0 when x = −1/3 So f is concave down on (−∞, −1/3), concave up on (−1/3, ∞), and has an inflection point at (−1/3, 2/27) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 20 / 30
  • 48. Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
  • 49. Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) = x − x = x (5x − 2) 9 9 9 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
  • 50. Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) =x − x = x (5x − 2) 9 9 9 The second derivative f′′ (x) is not defined at 0 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
  • 51. Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) =x − x = x (5x − 2) 9 9 9 The second derivative f′′ (x) is not defined at 0 Otherwise, x−4/3 is always positive, so the concavity is determined by the 5x − 2 factor . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
  • 52. Finding Intervals of Concavity II Example Find the intervals of concavity of the graph of f(x) = x2/3 (x + 2). Solution 10 −1/3 4 −4/3 2 −4/3 f′′ (x) =x − x = x (5x − 2) 9 9 9 The second derivative f′′ (x) is not defined at 0 Otherwise, x−4/3 is always positive, so the concavity is determined by the 5x − 2 factor So f is concave down on (−∞, 0], concave down on [0, 2/5), concave up on (2/5, ∞), and has an inflection point when x = 2/5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 21 / 30
  • 53. The Second Derivative Test Theorem (The Second Derivative Test) Let f, f′ , and f′′ be continuous on [a, b]. Let c be be a point in (a, b) with f′ (c) = 0. If f′′ (c) < 0, then c is a local maximum. If f′′ (c) > 0, then c is a local minimum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 22 / 30
  • 54. The Second Derivative Test Theorem (The Second Derivative Test) Let f, f′ , and f′′ be continuous on [a, b]. Let c be be a point in (a, b) with f′ (c) = 0. If f′′ (c) < 0, then c is a local maximum. If f′′ (c) > 0, then c is a local minimum. Remarks If f′′ (c) = 0, the second derivative test is inconclusive (this does not mean c is neither; we just don’t know yet). We look for zeroes of f′ and plug them into f′′ to determine if their f values are local extreme values. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 22 / 30
  • 55. Proof of the Second Derivative Test Proof. Suppose f′ (c) = 0 and f′′ (c) > 0. Since f′′ is continuous, f′′ (x) > 0 for all x sufficiently close to c. Since f′′ = (f′ )′ , we know f′ is increasing near c. Since f′ (c) = 0 and f′ is increasing, f′ (x) < 0 for x close to c and less than c, and f′ (x) > 0 for x close to c and more than c. This means f′ changes sign from negative to positive at c, which means (by the First Derivative Test) that f has a local minimum at c. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 23 / 30
  • 56. Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
  • 57. Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
  • 58. Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. Remember f′′ (x) = 6x + 2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
  • 59. Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. Remember f′′ (x) = 6x + 2 Since f′′ (−2/3) = −2 < 0, −2/3 is a local maximum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
  • 60. Using the Second Derivative Test I Example Find the local extrema of f(x) = x3 + x2 . Solution f′ (x) = 3x2 + 2x = x(3x + 2) is 0 when x = 0 or x = −2/3. Remember f′′ (x) = 6x + 2 Since f′′ (−2/3) = −2 < 0, −2/3 is a local maximum. Since f′′ (0) = 2 > 0, 0 is a local minimum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 24 / 30
  • 61. Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
  • 62. Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
  • 63. Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 10 −4/3 Remember f′′ (x) = x (5x − 2), which is negative when 9 x = −4/5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
  • 64. Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 10 −4/3 Remember f′′ (x) = x (5x − 2), which is negative when 9 x = −4/5 So x = −4/5 is a local maximum. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
  • 65. Using the Second Derivative Test II Example Find the local extrema of f(x) = x2/3 (x + 2) Solution 1 −1/3 Remember f′ (x) = x (5x + 4) which is zero when x = −4/5 3 10 −4/3 Remember f′′ (x) = x (5x − 2), which is negative when 9 x = −4/5 So x = −4/5 is a local maximum. Notice the Second Derivative Test doesn’t catch the local minimum x = 0 since f is not differentiable there. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 25 / 30
  • 66. Using the Second Derivative Test II: Graph Graph of f(x) = x2/3 (x + 2): y . . −4/5, 1.03413) ( . . . 2/5, 1.30292) ( . . x . . −2, 0) ( . 0, 0) ( . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 26 / 30
  • 67. When the second derivative is zero At inflection points c, if f′ is differentiable at c, then f′′ (c) = 0 Is it necessarily true, though? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 27 / 30
  • 68. When the second derivative is zero At inflection points c, if f′ is differentiable at c, then f′′ (c) = 0 Is it necessarily true, though? Consider these examples: f(x) = x4 g(x) = −x4 h(x) = x3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 27 / 30
  • 69. When first and second derivative are zero function derivatives graph type f′ (x) = 4x3 , f′ (0) = 0 f(x) = x4 min f′′ (x) = 12x2 , f′′ (0) = 0 . . g′ (x) = −4x3 , g′ (0) = 0 g(x) = −x4 max g′′ (x) = −12x2 , g′′ (0) = 0 h′ (x) = 3x2 , h′ (0) = 0 h(x) = x3 infl. h′′ (x) = 6x, h′′ (0) = 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 28 / 30
  • 70. When the second derivative is zero At inflection points c, if f′ is differentiable at c, then f′′ (c) = 0 Is it necessarily true, though? Consider these examples: f(x) = x4 g(x) = −x4 h(x) = x3 All of them have critical points at zero with a second derivative of zero. But the first has a local min at 0, the second has a local max at 0, and the third has an inflection point at 0. This is why we say 2DT has nothing to say when f′′ (c) = 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 29 / 30
  • 71. Summary Concepts: Mean Value Theorem, monotonicity, concavity Facts: derivatives can detect monotonicity and concavity Techniques for drawing curves: the Increasing/Decreasing Test and the Concavity Test Techniques for finding extrema: the First Derivative Test and the Second Derivative Test . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Shapes of Curves June 9, 2010 30 / 30