"Basic Inequalities" Quadratic Functions, Equations and Inequalities PPT (Basic Inequalities in the First Lesson)

Description

"Basic Inequalities" Quadratic Functions, Equations and Inequalities PPT (Basic Inequalities in the First Lesson)

Part One: Learning Objectives

Understand the content and derivation process of basic inequalities

Able to use basic inequalities to find the maximum value of a function or algebraic expression

Basic Inequalities PPT, Part 2: Independent Learning

Problem guide

Preview the textbook P44-P46 and think about the following questions:

1. What are the contents of the basic inequalities?

2. What are the conditions for the basic inequality to hold?

3. What issues should be paid attention to when using basic inequalities to find the optimal value?

A preliminary exploration of new knowledge

1. Important and basic inequalities

■Instructions from famous teachers

(1) The conditions for the establishment of the two inequalities a2+b2≥2ab and a+b2≥ab are different. The former requires that a and b be real numbers, while the latter requires that a and b are both positive real numbers (actually the latter only requires a≥0 and b≥0).

(2) The two inequalities a2+b2≥2ab and a+b2≥ab are both inequalities with an equal sign, and both are "if and only if a=b, the equal sign holds true".

2. Basic Inequalities and Maximum Values

It is known that x>0, y>0, then

(1) If x+y=S (the sum is a constant value), then when _______, the product xy obtains the maximum _______ value_______.

(2) If xy=P (the product is a constant value), then when _______, the sum x+y obtains the maximum _______ value_______.

Memory tips: The sum of two positive numbers must have the largest product, and the product of two positive numbers must have the smallest sum.

■Instructions from famous teachers

To use basic inequalities to find the optimal value, we must follow the principle of "one is positive, two are definite, and three are equal", that is:

①One positive: It meets the prerequisites for the establishment of the basic inequality a+b2≥ab, a>0, b>0;

②Second Definite: Convert one side of the inequality to a definite value;

③Three equals: The conditions for taking the "=" sign must exist, that is, the "=" sign is established.

The above three points are indispensable.

self-test

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

(1) For any a, b∈R, a2+b2≥2ab is true. ()

(2) If a>0, b>0 and a≠b, then a+b>2ab.()

(3) If a>0, b>0, then ab≤a+b22.()

(4) When a and b have the same sign, ba+ab≥2.()

(5) The minimum value of the function y=x+1x is 2.()

If a>0, then the minimum value of a+1a+2 is ()

A． 2B. twenty two

C. 3D. 4

Basic Inequalities PPT, Part 3: Interactive Lecture and Practice

Understanding basic inequalities

Which of the following conclusions is correct ()

A． If x∈R, and x≠0, then 4x+x≥4

B． When x>0, x+1x≥2

C. When x≥2, the minimum value of x+1x is 2

D. When 0

[Analysis] For option A, when x<0, 4x+x≥4 is obviously not true; for option B, it meets the three basic conditions for applying basic inequalities: "one is positive, two is definite, and three are equal"; for option C, verification is ignored The conditions for the establishment of the equal sign, that is, x=1x, then x=±1, do not satisfy x≥2; for option D, x-1x increases monotonically in the range of 0

The following conditions are given: ①ab>0; ②ab<0; ③a>0, b>0; ④a<0, b<0. Among them, the conditions that can make ba+ab≥2 hold are ()

A． 1 B. 2

C. 3 D. 4

Use basic inequalities to directly find the optimal value

(1) It is known that t>0, find the minimum value of y=t2-4t+1t;

(2) If positive real numbers x and y satisfy 2x+y=1, find the maximum value of xy.

regular method

(1) If a+b=S (the sum is a constant value), when a=b, the product ab has the maximum value S24, which can be obtained by using the basic inequality ab≤a+b2.

(2) If ab=P (the product is a constant value), then when a=b, the sum a+b has a minimum value 2P, which can be obtained by using the basic inequality a+b≥2ab.

In either case, attention must be paid to whether the conditions for obtaining the equal sign are established.

1. It is known that x>0, y>0, and x+y=8, then the maximum value of (1+x)(1+y) is ()

A． 16B. 25

C. 9 D. 36

2. If a and b are both positive numbers, then the minimum value of 1+ba1+4ab is ()

A． 7 b． 8

C. 9 D. 10

Find the optimal value using basic inequalities

(1) It is known that x>2, then the minimum value of y=x+4x-2 is ________.

(2) If 0

(3) If x, y∈(0,+∞), and x+4y=1, then the minimum value of 1x+1y is ________.

Solving strategy

A strategy for finding the optimal value using basic inequalities through the patchwork method

The essence of the patchwork method lies in the flexible deformation of algebraic expressions. Piecing together coefficients and constants is the key. When using the patchwork method to find the optimal value, attention should be paid to the following aspects:

(1) The skill of piecing together is based on the integer equation. Pay attention to the changes in coefficients and the adjustment of the constants in the equation to achieve equivalent deformation.

(2) The goal of deformation of algebraic expressions is to piece together the definite value of the sum or product.

(3) When removing terms and adding terms, attention should be paid to checking the premise of using basic inequalities.

Basic Inequalities PPT, Part 4: Feedback on Compliance

1. Among the following inequalities, the correct one is ()

A． a+4a≥4　B． a2+b2≥4ab

C. ab≥a+b2 D． x2+3x2≥23

2. If a>0, b>0, a+2b=5, then the maximum value of ab is ()

A.25 B.25/2

C.25/4 D.25/8

3. If a>1, then the minimum value of a+1a-1 is ()

A． 2B. a

C. 2aa-1 D． 3

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Update Time: 2024-10-03

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