3. Theme 3. GTLN-GTNN của hàm số (handout).pdf

NỀN TẢNG HÀM SỐ TOÁN 12
Giá trị lớn nhất – giá trị nhỏ nhất của hàm số
3
Buổi

2
Theme 3
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
1. Khái niệm giá trị lớn nhất, giá trị nhỏ nhất của hàm số f(x) trên D
Cho hàm số  
f x xác định trên tập D.
• Số M được gọi là giá trị lớn nhất của hàm số  
y f x
 trên tập D nếu  
f x M
 với mọi x D
 và tồn tại
0
x D
 sao cho  
0
.
f x M
 Kí hiệu  
max
x D
M f x

 hoặc  
max .
D
M f x

• Số m được gọi là giá trị nhỏ nhất của hàm số  
y f x
 trên tập D nếu  
f x m
 với mọi x D
 và tồn tại
0
x D
 sao cho  
0
.
f x m
 Kí hiệu  
min
x D
M f x

 hoặc  
min .
D
M f x

[Ví dụ 1] Tìm giá trị lớn nhất và nhỏ nhất của hàm số   2
4 .
y f x x
  
Lời giải
Điều kiện: 2
4 0 2 2.
x x
      Tập xác định 2;2 .
D  
 
 
Ta có:
   2
4 0;
f x x
   dấu bằng xảy ra khi 2 2
4 0 .
2
x
x
x
  
   


Do đó      
2;2
min 2 2 0.
f x f f
 
 
   
   2
4 4 2;
f x x
    dấu bằng xảy ra khi 2
4 4 0.
x x
    Do đó    
2;2
ma 0 2.
x f x f
 
 
 
REMARK 1 Trong trường hợp ta có sẵn đồ thị hàm số  
y f x

 Nếu xét trên ;
a b
 
  thì ta thấy trong tất cả các giá trị  
f x khi ;
x a b
 
  thì  
f  là tung độ y cao nhất
trên Oy ứng với ;
x a b
 
 
    ; .
f f x x a b
 
     
 Tương tự nếu xét trên ;
a b
 
  thì  
f  là tung độ thấp nhất trên Oy ứng với ;
x a b
 
 
    ; .
f f x x a b
 
     
O x
y
f(a)
f(α)
a b
α
y = f(x)
giá trị lớn nhất của hàm số
y = f(x)
giá trị nhỏ nhất của hàm số
y = f(x) trên [a;b]

3
Phần 2. Kiến thức nền tảng hàm số
Ví dụ 1 Cho hàm số  
y f x
 liên tục trên đoạn 2;6
 

  có đồ thị như hình
vẽ bên. Gọi M, m lần lượt là giá trị lớn nhất, giá trị nhỏ nhất của  
f x trên
đoạn 2;6
 

  . Giá trị của 2 3
M m
 là bao nhiêu?
A. max 10.
y 
B. max 4.
y 
C. max 2.
y 
D. max 2 5.
y 
y f x
 có đồ thị như hình vẽ bên.
Giả sử kí hiệu    
1;5
1;5
max ; min .
M f x m f x
 
   
 
  Tìm 2 3 .
M m

A. 8.
B. 11.
C. 9.
D. 6.
O x
y
6
5
2
1
1
4
6
-1
-2
-4
O x
y
1 2 3 4 5
1
2
3
4
5
6
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4
Theme 3
REMARK 2 Để tìm GTLN, GTNN của hàm số  
f x trên một khoảng, đoạn hay nửa khoảng, ta lập bảng
biến thiên của hàm số trên tập hợp đó.
[Ví dụ 2] Giải ví dụ 1 bằng cách lập bảng biến thiên.
Lời giải
 Bước 1: Tìm nghiệm của 0.
y 
Với  
2;2 ,
x  ta có:
 
2
2 2
4
; 0 0.
2 4 4
x x
y y x
x x
 


     
 
 Bước 2: Lập bảng biến thiên của hàm số trên đoạn 2;2 .
 

 
 Bước 3: Kết luận.
Từ bảng biến thiên, ta được:          
2;2 2;2
min 2 2 0; max 0 2.
f x f f f x f
   
   
     
–2 0 2
+ –
0
2
0
x
y
y'
0

5
2. Các bước tìm giá trị lớn nhất – giá trị nhỏ nhất của hàm số bằng đạo hàm
Tìm GTLN- GTNN trên đoạn ;
a b
 
  Tìm GTLN – GTNN trên khoảng  
;
a b
Xét hàm số  
y f x
 liên tục trên đoạn ;
a b
 
  và có
đạo hàm trên khoảng  
,
a b , có thể trừ một số hữu
hạn điểm.
Nếu   0
f x
  chỉ tại một số hữu hạn điểm thuộc
khoảng  
;
a b thì ta có quy tắc sau
Quy tắc tìm GLTN, GTNN trên đoạn
y f x
 liên tục trên ;
a b
 
  .
1. Tìm các điểm 1 2
; ;...; n
x x x trên khoảng  
; ,
a b tại
đó  
f x
 bằng 0 hoặc  
f x
 không xác định (gọi
là các điểm tới hạn).
2. Tính          
1 2
; ; ;...; ; .
n
f a f x f x f x f b
3. Tìm số lớn nhất M và số nhỏ nhất m trong các
số trên. Ta có    
;
;
max ; min .
a b
a b
M f x m f x
 
   
 
 
Trong trường hợp đề yêu cầu tìm GTLN, GTNN trên
một khoảng  
;
a b thì ta thực hiện vẽ bảng biến thiên
của hàm số, rồi từ đó so sánh và xác định GTLN và
GTNN.
Quy tắc tìm GLTN, GTNN trên khoảng
y f x
 liên tục trên  
;
a b .
Vẽ bảng biến thiên của hàm số trên  
; .
a b
1. Tìm các điểm 1 2
; ;...; n
; ,
a b tại đó
 
f x
 bằng 0 hoặc  
f x
 không xác định (gọi là các
điểm tới hạn).
2. Vẽ bảng biến thiên trên  
; .
a b
Điền các điểm 1 2
; ;...; n
;
a b vào
bảng biến thiên, xét dấu đạo hàm, và vẽ bảng biến
thiên.
3. Nhìn vào bảng biến thiên kết luận.
REMARK 1 Mọi hàm số liên tục trên một đoạn đều có giá trị nhỏ nhất và giá trị lớn nhất trên đoạn đó.
REMARK 2 Nếu hàm số  
y f x
 xác định trên ;
a b
 
  và luôn đồng biến (hoặc luôn nghịch biến) trên  
;
a b
thì hàm số đạt giá trị lớn nhất, giá trị nhỏ nhất tại các điểm mút của đoạn ;
a b
 
  .
Cụ thể:
 
f x đồng biến trên  
;
a b  
f x nghịch biến trên  
;
a b
x
f’(x)
f(x)
a b
f(a)
f(b)
min
a;b
f(x)
max
a;b
f(x)
+
x
f’(x)
f(x)
a b
f(a)
f(b)
min
a;b
f(x)
max
a;b
f(x)
–

6
Theme 3
Dạng thức 1 Câu trắc nghiệm nhiều phương án lựa chọn. Mỗi câu hỏi thí sinh chỉ chọn một phương án.
y f x
 có đồ thị như hình vẽ bên dưới, biết  
1;2
max ;
M f x
 
 

 
1;0
min .
m f x
 
 
 Tìm .
M m

A. 1.
B. 0.
C. 5.
D. 3.
y f x
 liên tục và xác định trên
3;2
 

  và có bảng biến thiên như hình vẽ. Gọi M và m lần
lượt là giá trị lớn nhất và giá trị nhỏ nhất của hàm số
 
y f x
 trên 1;2 .
 

  Giá trị của M m
 bằng bao nhiêu?
A. 3. B. 1.
C. 2. D. 0.
Ví dụ 3 Tìm giá trị lớn nhất M, giá trị nhỏ nhất m của hàm số 4 2
2 1
y x x
   trên 1; 3 .
 
 
A. 3; 1.
M m
  B. 14; 2.
M m
  C. 14; 3.
M m
  D. 2; 1.
M m
 
O x
y
2
1
3
3
-1
2
-2
0
0
1
0
x
f’(x)
–1
–3 2
+
+ _ _
0
f (x)
3 2
0
–2 1
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7
Ví dụ 4 Giá trị lớn nhất của hàm số 3 5 7 3
y x x
    là
A. max 10.
y  B. max 4.
y  C. max 2.
y  D. max 2 5.
y 
Ví dụ 5 (đề tham khảo năm 2024)
Giá trị lớn nhất của hàm số   4 2
6 4
f x x x
    bằng
A. 3.
 B. 4.
 C. 5. D. 3.
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8
Theme 3
y f x
 xác định trên  
5;5
 có
bảng biến thiên như hình vẽ. Phát biểu nào sau đây là đúng?
A.
   
5;5 5;5
min 10; max 10.
y y
 
  
B.
   
5;5 5;5
min 5; max 10.
y y
 
  
C.
   
5;5 5;5
min 5; max 0.
y y
 
  
D. Hàm số không có GTLN, GTNN trên  
5;5 .

Ví dụ 7 (đề tham khảo năm 2017)
Tính giá trị nhỏ nhất của hàm số 2
4
3
y x
x
  trên khoảng  
0; .
A.
 
0;
33
min
5
y

 . B.
 
3
0;
min 2 9
y

 . C.
 
3
0;
min 3 9
y

 . D.
 
0;
min 7
y

 .
Ví dụ 8 Giá trị nhỏ nhất của hàm số 4 3
y x
   trên tập xác định của nó là
A. 2 3.
 B. 2 3. C. 0. D. 3.
x
f’(x)
–5
f (x)
10
–2 0
+
+
–5
0 –
0
5
–10
0
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9
Dạng thức 2 Câu trắc nghiệm đúng sai. Trong mỗi ý a), b), c), d) thí sinh chọn đúng hoặc sai.
Ví dụ 9 Cho hàm số 2
2 3.
y x x
   
a) Giá trị lớn nhất của hàm số trên 1;3
 

  bằng 2.
b) Giá trị nhỏ nhất của hàm số đã cho bằng 0.
c) Giá trị lớn nhất của hàm số đã cho bằng 2.
d) Hàm số đạt giá trị lớn nhất tại 1.
x  
Ví dụ 10 Cho hàm số   3
2 5
x
f x
x



trên 0;2 .
 
 
a) Giá trị lớn nhất của  
f x bằng 0,6.
b) Giá trị nhỏ nhất của  
f x bằng
5
.
9
c) Giá trị lớn nhất của  
f x đạt được tại 2.
x 
d) Giá trị nhỏ nhất của  
f x đạt được tại 1.
x 
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