"Basic Inequalities" Quadratic functions, equations and inequalities PPT

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"Basic Inequalities" Quadratic functions, equations and inequalities PPT

Part One: Explanation of Curriculum Standards

1. Understand the basic inequality √ab≤(a+b)/2(a,b≥0).

2. Be able to use basic inequalities to solve simple maximum or minimum problems.

3. Be able to use basic inequalities to prove inequalities and compare the magnitude of algebraic expressions.

Basic Inequalities PPT, Part 2: Independent Learning

1. Basic Inequalities

1. (1) In the last lesson, we learned an important inequality: if a, b∈R, then a2+b2≥2ab (the equal sign holds true if and only if a=b). If a>0 ,b>0, we replace a and b in the inequality with √a and √b respectively. What form can we get?

Hint: get a+b≥2√ab.

(2) We often write this inequality as √ab≤(a+b)/2(a>0,b>0), and call this inequality "basic inequality". What are the conditions for the equality sign to hold?

Tip: The equal sign holds true if and only if a=b.

(3) We call √ab the geometric mean of a and b, and (a+b)/2 the arithmetic mean of a and b. How to use these two concepts to describe basic inequalities?

Tip: The basic inequality can be stated as: the arithmetic mean of two positive numbers is not less than their geometric mean.

(4) As shown in the figure, AB is the diameter of the circle, point C is a point on AB, AC=a, BC=b. Draw a chord DD' perpendicular to AB through point C, connecting AD and BD.

2. Use basic inequalities to find the optimal value

1. Fill out the two forms below:

Based on the above table and combined with basic inequality analysis:

(1) When x+y is a constant value, does xy have a maximum value or a minimum value? What is the maximum value equal to?

(2) When xy is a constant value, does x+y have a maximum value or a minimum value? What is the maximum value equal to?

Tips: Fill out the table. (1) When x+y is a fixed value, xy has a maximum value, and the maximum value is equal to

2. Fill in the blanks

Basic Inequalities and Maximum Values

It is known that x and y are both positive numbers.

3. Do it

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

(1) If xy=4, then the minimum value of x+y is___________;

(2) If x+y=4, then the maximum value of xy is___________.

Basic Inequalities PPT, Part 3: Inquiry and Learning

Understanding basic inequalities

Example 1 Which of the following propositions is correct ()

Answer:B

Reflection: Pay attention to the following three points when applying basic inequalities.

(1) Each term or factor is positive;

(2) The sum or product is a constant value;

(3) Each term or factor can obtain equal values. That is, "one positive, two definite three are equal".

Variation Training 1 Which of the following conclusions is not true ()

A. If a,b∈R, then a10+b10≥2a5b5

D. If a∈R, then a2+9≥6a

Answer:C

Exploration 2 Use basic inequalities to prove inequalities

Example 2(1) It is known that a, b, c are not all equal positive real numbers,

Prove: a+b+c>√ab+√bc+√ca.

(2) It is known that a, b, c are positive real numbers, and a+b+c=1,

Verify: (1/a "-" 1)(1/b "-" 1)(1/c "-" 1)≥8.

Analysis: (1) The left side of the inequality is the sum, and the right side is the sum of the product expressions with the root sign. Use basic inequalities to convert the sum into the product, and prove the inequality. (2) The number on the right side of the inequality is 8, which reminds us By using basic inequalities for the factors on the left, we can get three consecutive multiplications of "2";

Basic Inequalities PPT, Part 4: Practice in class

1. It is known that x>0, then the minimum value of x+1/2x is ()

A.1/2 B.1 C.√2/2 D.√2

Analysis: From the basic inequality, we get x+1/2x≥2√(x"•" 1/2x)=√2. The equal sign holds true if and only if x=1/2x, that is, x=√2/2. Therefore, the minimum value is √2.

Answer:D

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Update Time: 2024-06-17

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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