"Basic Properties of Functions" Concept and Properties of Functions PPT (Maximum and Minimum Values ​​of Functions in Lesson 2)

Description

"Basic Properties of Functions" Concept and Properties of Functions PPT (Maximum and Minimum Values ​​of Functions in Lesson 2)

Part One: Learning Objectives

Understand the maximum (minimum) value of a function and its geometric meaning, and be able to find the maximum (minimum) value of a function with the help of images

Will use the monotonicity of the function to find the optimal value

Ability to use the maximum value of a function to solve relevant simple practical problems

Basic properties of functions PPT, part 2: independent learning

Problem guide

Preview textbooks P79-P81 and think about the following questions:

1. As can be seen from the graph of the function, what is the geometric significance of the maximum (small) value of the function?

2. What are the definitions of the maximum and minimum values ​​of a function?

A preliminary exploration of new knowledge

1. maximum value of function

Generally speaking, assuming the domain of function y=f(x) is I, if there is a real number M that satisfies:

(1)∀x∈I, all have __________;

(2)∃x0∈I, such that __________.

Then, we say M is the maximum value of function y=f(x).

2. Minimum value of function

Generally speaking, assuming the domain of function y=f(x) is I, if there is a real number M that satisfies:

(1)∀x∈I, all have __________;

(2)∃x0∈I, such that __________.

Then, we say that M is the minimum value of function y=f(x).

■Instructions from famous teachers

Two keywords in the definition of function maximum and minimum

(1)∃(existence)

M is first a function value, which is an element in the value range. For example, the minimum value of function y=x2 (x∈R) is 0, and f(0)=0.

(2)∀(any)

The ∀ (arbitrary) in the definition of the maximum (small) value means that every value in the domain must satisfy an inequality, that is, for all elements in the domain, f(x)≤M(f(x)≥M) It is established, that is to say, the graph of function y=f(x) cannot be located above (below) the straight line y=M.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) Any function has a maximum or minimum value. ()

(2) The minimum value of the function must be smaller than the maximum value. ()

(3) If the function f(x)≤1 is always true, then the maximum value of f(x) is 1.()

The graph of function f(x) on [-2,2] is as shown in the figure, then the minimum value and maximum value of this function are ()

A． -1,0 B． 0, 2

C. -1,2 D． 12,2

Function f(x)=1x on [1,+∞)()

A． There is a maximum value but no minimum value

B． There is a minimum value but no maximum value

C. There is a maximum value and a minimum value

D. No maximum or minimum value

Basic properties of functions PPT, the third part: interactive teaching and practice

Image method to find the maximum value of a function

It is known that the function f(x)=-2x, x∈(-∞,0), x2+2x-1, x∈[0,+∞).

(1) Draw the graph of the function and write the monotonic interval of the function;

(2) Find the minimum value of the function based on the graph of the function.

[Solution] (1) The graph of the function is as shown in the figure.

It can be seen from the image that the monotonically increasing intervals of f(x) are (-∞, 0) and [0, +∞), and there is no decreasing interval.

(2) It can be seen from the function graph that the minimum value of the function is f(0)=-1.

regular method

General steps for finding the optimal value using the image method

Track training

1. The graph of function f(x) on the interval [-2,5] is as shown in the figure, then the minimum value and maximum value of this function are ()

A． -2,f(2)　B． 2,f(2)

C. -2, f(5) D. 2,f(5)

2. It is known that the function f(x)=x2-x (0≤x≤2), 2x-1 (x>2), find the maximum and minimum values ​​of the function f(x).

Solution: Draw the image of f(x) as shown in the figure. It can be seen from the image that when x=2, f(x) takes a maximum value of 2;

When x=12, the minimum value of f(x) is -14.

So the maximum value of f(x) is 2 and the minimum value is -14.

Use the monotonicity of the function to find the optimal value

It is known that the function f(x)=x-1x+2, x∈[3, 5].

(1) Determine the monotonicity of function f(x) and prove it;

(2) Find the maximum and minimum values ​​of function f(x).

regular method

The relationship between the maximum value of a function and monotonicity

(1) If the function f(x) is a decreasing function on the closed interval [a, b], then the maximum value of f(x) on [a, b] is f( a), the minimum value is f(b).

(2) If the function f(x) is an increasing function on the closed interval [a, b], then the maximum value of f(x) on [a, b] is f( b), the minimum value is f(a).

[Note] When finding the maximum value, you must pay attention to the opening and closing of the given interval. If it is an open interval, there may not be a maximum value.

Basic properties of functions PPT, Part 4: Feedback on achievement of standards

1. The graph of function f(x) is as shown in the figure, then its maximum value and minimum value are () respectively

A． f32, f-32 B. f(0),f32

C. f-32, f(0) D. f(0), f(3)

2. Suppose the function f(x)=x|x| defined on R, then f(x)()

A． only maximum value

B． only minimum value

C. There is both a maximum value and a minimum value

D. There is neither maximum nor minimum value

3. If the minimum value of function f(x)=1x on [1, b](b>1) is 14, then b=________.

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Update Time: 2024-07-08

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