Solve problem 10 of the mean inequality

Solution: y=x+3 (x-2).

y=(x-2)+3/(x-2)+2

Obtained by a+b 2 ab.

y 2 (x-2)*3 (x-2) +2=2( 3+1) so y(min)=2( 3+1).

When x-2=3 (x-2), the minimum value, i.e., x-4x+1=0, is obtained, and x=2 3 is obtained

Because x>2, x=2+ 3

Note: Indicates the root number.

I studied science in high school and was proficient in mathematics. In fact, the key to the problem of mean inequality is to find a and b and apply the formula. Do a few more practice questions and you'll be handy.

y=x+(3/（x-2)）

x+(3/（x-2))-2+2

x-2)+(3/（x-2))+2

Because x>2, x-2>0

It is obtained from the mean inequality.

x-2)+(3 (x-2))>=2*(x-2)) (3 (x-2)).

2 * root number 3 so the minimum value of y = 2 * root number 3 + 2

Combining y=2*root, number, 3+2 generations, the original formula yields the x value ......!

y=x+3 (x-2)=(x-2)+3 (x-2)+2>=2*root number 3+2

The minimum value is 2 * root number 3 + 2 and the corresponding value of x is 2 + root number 3

The derivative is first found, and then it is equal to zero to find the extreme value, and after analysis, it is found that the minimum is taken when x=2.

The common question types and analysis of mean inequalities are as follows:

1. If a, b, and c are unequal real numbers, verify a2+b2+c2>ab+bc+ac. Proof:

a, b, c are unequal real numbers. ∴ a2+b2>2ab， a2+c2>2ac， b2+c2>2bc。The above three equations are added together to give 2a2+2b2+2c2>ab+2bc+2ac.

i.e. A2+B2+C2>AB+BC+AC.

2. Both refer to the basic properties of silver value inequality.

1. If x>y, then yy (symmetry).

2. If x>y, y>z; Then x>z (transitivity).

3. If x>y, and Kaiqing z is an arbitrary real number or integer, then x+z>y+z (addition principle, or co-directional inequality is only additive).

4. If x>y, z>0, then xz>yz; If x>y,z<0, then xz<>

5. If x>y, m>n, then x+m>y+n (sufficient and unnecessary).

Mean inequality, also known as mean inequality and average inequality, is an important formula in mathematics. The content of the formula is hn gn an qn, that is, the harmonic mean does not exceed the geometric mean, the geometric mean does not exceed the arithmetic mean, and the arithmetic mean does not exceed the square mean.

<> factor 1-x 2 with the squared difference formula, and then use the mean inequality directly to find the maximum value of 1.

Hehe. That's how it came about.

Because (a-b) 2=a 2-2ab+b 2 0 collation yields a 2+b 2 2ab

Obviously. Only if a=b

Only then can there be (a-b) 2=0

So. If and only if a=b takes an equal sign.

And because when A≠B.

ab cannot be 0 at the same time

Because A2+B22AB must be satisfied

Therefore, it is necessary that both A 2 and B 2 are "0".

Therefore, it is commuted t=a 2>0, u=b 2>0

It is T+U 2 root number under (tu).

If and only if t=u is equal to the equal.

This is the mean inequality.

Your answer is correct, but there are flaws in the answering process

When x 0, x+(9 4x) 0, to use the mean theorem, we must ensure x 0, combined with this problem, we can consider finding the maximum value of the opposite number of this equation first, and then finding the maximum value of the formula.

x)+[9/(－4x)]

2 root number [( x) 9 ( 4x)].

2 root number [9 4].

So, x+[9 (4x)] 3

That is, the maximum value of the algebraic formula is 3.

The result is right, the process is wrong.

The first line, followed by the second equal sign, and after the less than or equal sign, add a negative sign each.

In the second line, the minus sign in front of (-x) is thrown outside the middle parentheses.

In the third line, the minus sign in front of (-x) is also thrown outside the middle parentheses.

Because a is in (0,1), then b>1, then a+2 a 2 root number 2, at this time a = 2 a, a = root number 2, when limiting the range, consider whether the mean inequality is taken, only according to the monotonicity of the function to solve, so a + 2 a is a subtraction function, that is, to obtain (3, positive infinity).

Didn't the title say, ignore A?

Answer: Make use of mean inequalities.

a²+b²≥2ab

2（a²+b²）≥a²+b²+2ab=(a+b)²∴2x+1)+√2y+1)]²2[√(2x+1)²+2x+1)²]

i.e. [ 2x+1)+ 2y+1)] 2(2x+1+2y+1)=8

(2x+1)+ 2y+1) 2 2 if and only if (2x+1)= (2y+1), i.e. x=y=1 2 the equal sign holds.

(2x+1)+2y+1) is 2 2

Let the maximum value be a, how can a 2=(2x+1)+(2y+1)+2 root number [(2x+1)(2y+1)]=2(x+y+1)+2 root number [4xy+2(x+y)+1].

Bring x+y=1 in, and a 2=4+2 (4xy+3). In this case, the mean theorem is used, 4xy is less than or equal to (x+y) 2, that is, 4xy is less than or equal to 1, because a is the maximum value, so 4xy=1. then a 2 = 4 + 4 = 8. Since a is the maximum, a = 2 times the root 2

Just use this. Cauchy inequalities are also possible.

Construct it. However, the interval given in the question is an open interval, so there is no minimum value.

y=(x+1)+1/(x+1)-1

According to the mean inequality a+b 2ab

So (x+1)+1 (x+1) 2

So y should be 2-1=1