The problem of finding the maximum value of the basic inequality, and finding the maximum value of t

Since x,y is known to be positive real numbers, and x+y=1, then.

x²/(x+2)+y²/(y+1)

x²-4+4)/(x+2)+(y²-1+1)/(y+1)x-2)+(y-1)+4/(x+2)+1/(y+1)4/(x+2)+1/(y+1)-2

4/(x+2)+1/(2-x)-2

10-3x)/(4-x²)-2

Establish. f(x)=(10-3x) (4-x). f'(x) = (-3x +20x-12) (4-x ) let f'(x)=0.

x=2 3 (or 6 rounding).

Old. 0,2 3) is the monotonic decreasing interval, and (2 3,1) is the monotonic increasing interval.

Namely. When x=2 3, f(x) takes the minimum value.

At this time. x (x+2) + y (y+1) = f(x)-2 also obtains a minimum value.

The calculation yields the minimum value = 1 4

1. Pay attention to the conditions that the fundamental theorem should meet.

Fundamental inequalities have the function of converting "sum" into "product" and "product" into "sum", but one.

We must pay attention to the premise of application: "one positive", "two fixed", "three equal" The so-called "one positive" refers to "positive number", "two fixed" refers to the application of the theorem to find the maximum value, the sum or product is a fixed value, and "three equal" refers to the condition that the equal sign is satisfied

Two. In conjunction with basic inequalities, it is necessary to pay attention to the fact that the conditions for its establishment should be consistent.

Some problems require multiple attempts to find the final result by using the fundamental inequality several times, and in this case, it is important to remember that the conditions for the equality sign to be true should be consistent when using this theorem continuously

For some problems, the basic inequality is directly used to find the maximum value, which does not meet the application conditions, but it can be used by adding terms, separating constants, squares, etc., so that the basic inequality can be used

1 Addition. 2 separation constants.

3 square.

Finding the Maximum Value of Fundamental InequalitiesThree Principles of Finding the Maximum Value of Fundamental Inequalities a, b are non-negative real numbers;

When and a+b is a fixed value, the product ab has a maximum value; When the product ab is a fixed value, and a+b have a minimum value;

When a=b, the equal sign in the inequality holds, and when a≠b, the equal sign in the inequality does not hold (in this case, a+b>2ab, which means that neither the minimum value of a+b nor the maximum value of ab exists).

A common deformation formula for fundamental inequalities.

1)ab≤(a,b)(a、ber);

2)ab≤ a2+b2 (a、ber);

3)(a+b)²≤2(a+b²)(a、ber).

Make up the strategy of "making up a fixed value".

The key to using the basic inequality to find the maximum value lies in how to make up the fixed value, which can be solved by using the deformation strategies such as the make-up term, the make-up coefficient, the overall substitution, the separation, the elimination element, the exchange element, the square, the structural inequality, the parameter method, the undetermined coefficient method, the homogeneity, the discriminant method, and the deflation.

Fundamental inequalities to find the maximum value method:Create the conditions for the establishment of the basic inequality:

One: all are positive;

2: The sum is a fixed value or the product is a fixed value;

Three: Two numbers are equal.

Abbreviation: one positive, two fixed, three equal.

"One positive" means that both formulas are positive, "two definite" means that when the basic inequality is applied to find the maximum value, the sum or product is a fixed value, and "three equal" means that the equal sign can be taken if and only if the two formulas are equal.

Two major techniques for solving fundamental inequalities:

1. The wonderful use of "1". If the sum of two formulas is a constant, the minimum value of the sum of the reciprocals of these two formulas is required, usually multiply the formula by 1, and then express 1 with the previous constant, and calculate the two formulas. If the sum of the reciprocals of the two formulas is known to be constant, find the minimum value of the sum of the two formulas, and the method is the same as above.

2. Adjustment factor. Sometimes when solving the maximum value of the product of two equations, the sum of the two equations needs to be constant, but many times it is not constant, and some of the coefficients need to be adjusted so that the sum is constant.

The conditions and steps for finding the maximum value using the basic inequality are as follows:

1. Create the conditions for the establishment of basic inequalities: all are positive; The sum is a fixed value or the product is a fixed value; The two numbers are equal.

Abbreviation: one positive, two fixed, three equal.

a+b 2 ab (a>0, b>0, equal sign when a and b are equal).

a2+b2 2ab(a2>0,b2>0,a2=b2).

2. The example questions are as follows:

When I got this question, some students began to use the basic inequality and think about the three conditions. x and y are both greater than 0, and the sum of x and 2y is a fixed value, and when these two numbers are equal, the maximum value of the product is found by the basic inequality, and then the maximum value of the denominator is obtained. But the molecule can't find it yet, and it can't do it blindly.

When you get a problem, when you can't get the maximum value of a basic inequality in one step, don't take it for granted that you are satisfied. At this time, you can simplify it first and observe it further.

If we want to get the minimum value and the product must be certain, then we will create the product certain.

Split the formula we got into two.

At this time, it is obvious that the multiplication of two numbers is a fixed value, and the root number xy is also a positive number, and the basic inequality can finally be obtained by xy=3.

3. Common methods for finding the maximum value.

1. Conventional matching method.

2. The substitution method of "1".

3., commutation method.

4. Multiplication and division coefficient method.

5. Elimination method (necessary constructor to find differences).

Finding the Maximum Value of Fundamental InequalitiesThree Principles of Finding the Maximum Value of Fundamental Inequalities a, b are non-negative real numbers;

When and a+b is a fixed value, the product ab has a maximum value; When the product ab is a fixed value, there is a minimum tremor hole with a+b;

A=B, the equal sign in the inequality holds Helu, and when a≠b, the equal sign in the inequality does not hold (in this case, a+b>2ab, which means that neither the minimum value of a+b nor the maximum value of ab exists).

A common deformation formula for fundamental inequalities.

1)ab≤(a,b)(a、ber);

2)ab≤ a2+b2 (a、ber);

3)(a+b)²≤2(a+b²)(a、ber).

Make up the strategy of "making up a fixed value".

The key to using the basic inequality to find the maximum value lies in how to make up the fixed value, which can be solved by using the deformation strategies such as the make-up term, the make-up coefficient, the overall substitution, the separation, the elimination element, the change of the eggplant, the square, the structural inequality, the parameter method, the undetermined coefficient method, the homogeneity, the discriminant method, and the deflation.

Since x,y is known to be positive real numbers, and x+y=1, then.

x²/(x+2)+y²/(y+1)

x²-4+4)/(x+2)+(y²-1+1)/(y+1)= (x-2)+(y-1)+4/(x+2)+1/(y+1)= 4/(x+2)+1/(y+1)-2

4/(x+2)+1/(2-x)-2

10-3x)/(4-x²)-2

Let f(x)=(10-3x) (4-x).

Derivation f'(x) = (-3x +20x-12) (4-x ) let f'(x)=0.

x=2 3 (or 6 rounding).

Therefore, (0,2 3) is the monotonic decreasing interval, and (2 3,1) is the monotonic increasing interval, that is, when x=2 3, f(x) is the minimum value.

In this case, x (x+2)+y (y+1)=f(x)-2 also obtains the minimum value, and the minimum value can be calculated to obtain the minimum value = 1 4

First passed.

ab+bc+ca)/abc=1/abc

Only the scope of the ABC is requested. By arithmetic greater than or equal to geometric average:

Cubic (ab*bc*ca)<=(ab+bc+ca) 3(ab*bc*ca)<=[(ab+bc+ca) 3] 3(abc) 2<=1 27

abc<=√3/9

.1/abc=>3√3

[Note: Here, there may be one less condition, that is, a, b, and c are all positive numbers.] Solution: ab+bc+ca=1and abc 0

Both sides are divided by ABC.

1/a)+(1/b)+（1/c)=1/(abc).

It is derived from the basic inequality.

1 a) + (1 b) + (1 c)] 27 (abc) Combined with the above results, it can be obtained.

1 a) + (1 b) + (1 c)] 27 (1 a) + (1 b) + (1 c) 3 3 equal sign is obtained only when a=b=c=1 3.

(1 a) + (1 b) + (1 c)] min = 3 3 is easy to know, and there is no maximum value in the original formula.