Bank soalan: Percubaan matematik tambahan kelantan 2013Ayu Lil'princess
This document contains a 25 question Additional Mathematics trial exam for the SPM 2013 examination. The exam consists of multiple choice and short answer questions testing concepts in arithmetic progressions, geometric progressions, functions, trigonometry, probability, and statistics. The student is expected to show working and steps to receive partial credit on questions.
This document contains a 25 question Additional Mathematics trial exam for the SPM 2013 examination. The exam consists of multiple choice and short answer questions testing concepts in arithmetic progressions, geometric progressions, functions, trigonometry, probability, and statistics. The student is expected to show working to receive partial credit on questions.
This document covers representing functions including objectives like defining functions and related terms, determining if a relation is a function, defining piecewise functions, and representing real-life situations
This document covers representing functions including objectives like defining functions and related terms, determining if a relation is a function, defining piecewise functions, and representing real-life situations
JEE Mathematics/ Lakshmikanta Satapathy/ Application of Derivative QA part 5/ Question on Absolute maximum and Minimum values of a function in a closed interval solved with the related concepts
To evaluate a function, substitute the input variable into the function in place of the original variable. This gives the output or image of the input under the function. For example, to evaluate f(x)=3x-5 at x=4, substitute 4 for x: f(4)=3(4)-5=7. When substituting expressions, use parentheses to avoid errors. For a function like f(x)=x^2-x-4, if f(k)=8, set the function equal to 8 and solve for k, finding that k=4 or k=-3.
This document contains notes from a calculus class covering topics including: implicit differentiation, related rates, linear approximations, maximum and minimum values, the mean value theorem, limits at infinity, and curve sketching. Example problems are provided for each topic to demonstrate key concepts and techniques.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
Lesson 2 - Functions and their Graphs - NOTES.pptJaysonMagalong
The document provides lesson material on functions and their graphs. It includes sections on defining functions, determining if a relation is a function, functional notation, domain and range, graphing functions, and identifying intervals of increase/decrease. Additional topics covered are relative min/max values, step functions, even and odd functions, and piecewise-defined functions. Examples and exercises are provided to illustrate key concepts.
This module introduces quadratic functions. It discusses identifying quadratic functions as those with the highest exponent of 2, rewriting quadratic functions in general form f(x) = ax^2 + bx + c and standard form f(x) = a(x-h)^2 + k, and the key properties of quadratic graphs including the vertex and axis of symmetry. The module provides examples of identifying quadratic functions from equations and ordered pairs/tables and rewriting quadratic functions between general and standard form using completing the square.
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
The document discusses several theorems related to polynomial functions:
1) The Remainder Theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a).
2) The Factor Theorem states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0.
3) The Rational Zero Theorem provides conditions for rational zeros of a polynomial with integral coefficients.
4) The Fundamental Theorem of Algebra states that every polynomial function with complex coefficients has at least one zero in the set of complex numbers.
The document defines relative and absolute maxima and minima for functions z=f(x,y). A relative maximum occurs when f(a,b) is greater than f(x,y) within a neighborhood circle of (a,b). An absolute maximum occurs when f(a,b) is greater than f(x,y) over the entire domain. Similarly for minima with the inequalities reversed. Extrema refer to maxima and minima. For a continuous function over a closed and bounded domain, absolute extrema exist and occur either in the interior or on the boundary. Examples find and classify extrema of functions.
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
This document discusses the Lagrange multiplier method for finding the constrained maximum or minimum of a function subject to an equality constraint. It provides examples of using Lagrange multipliers to find the dimensions of a rectangle with maximum area given a perimeter, and to find the points on a circle closest to and farthest from a given point. The key steps are to set up the Lagrange multiplier equation relating the gradients of the objective function and constraint, solve for the critical points, and evaluate the objective function at these points to find the maximum or minimum.
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3. Theme 3. GTLN-GTNN của hàm số (handout).pdf
1. NỀN TẢNG HÀM SỐ TOÁN 12
Giá trị lớn nhất – giá trị nhỏ nhất của hàm số
3
Buổi
2. 2
Theme 3
Giá trị lớn nhất – giá trị nhỏ nhất của hàm số
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. 3
Phần 2. Kiến thức nền tảng hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
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
Ví dụ 2 Cho hàm số
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. 4
Theme 3
Giá trị lớn nhất – giá trị nhỏ nhất của hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
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. 5
Phần 2. Kiến thức nền tảng hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
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
Xét hàm số
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
Xét hàm số
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
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. Vẽ bảng biến thiên trên
; .
a b
Điền các điểm 1 2
; ;...; n
x x x trên khoảng
;
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. 6
Theme 3
Giá trị lớn nhất – giá trị nhỏ nhất của hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
Dạng thức 1 Câu trắc nghiệm nhiều phương án lựa chọn. Mỗi câu hỏi thí sinh chỉ chọn một phương án.
Ví dụ 1 Cho hàm số
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.
Ví dụ 2 Cho hàm số
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. 7
Phần 2. Kiến thức nền tảng hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
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. 8
Theme 3
Giá trị lớn nhất – giá trị nhỏ nhất của hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
Ví dụ 6 Cho hàm số
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. 9
Phần 2. Kiến thức nền tảng hàm số
Group học |2K7 Xuất Phát Sớm | Quyết Đỗ Nguyện Vọng I 2025 🍀 | fb.com/groups/boner2k7
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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