The inequality puzzle, the master advances

Note that the upstairs is the maximum, why, the Cauchy inequality can be proved:

By Cauchy's inequality:

x+y<= (1 1+1 2)* x+y)=(2)* x+y) so it should be:

x+√y)/√(x+y)<=(√2)*√x+y)/√(x+y)=√2

2 is the maximum.

The minimum value should be taken when x=0, y=1 is alive x=1, y=0, the minimum value is 1 can be simply understood in this way, when finding the maximum value, the equal sign is established when the condition is x=y, that is, x, y is the closest to take the maximum value, of course, its opposite is that the greater the difference between x and y, the smaller its value, obviously, if x is infinite, y tends to 0 when the minimum is the same: this is equivalent to finding the limit:

lim(√x+√y)/√(x+y)=lim√x/√x=1。

The minimum value is 1, but this minimum value cannot be taken because of x,y r+,.

(x)+ y)] 2 [ (x+y)] 2x+y+2 (xy) x+y, first find the minimum value x+y+2 (xy) x+y] 1 2 (xy) x+yx+y, greater than or equal to 2 xy, the basic inequality.

Therefore, the minimum value of [ (x)+ y)] 2 [ (x+y)] 2 is 2 (x)+ y)] and the minimum value of [ (x+y)] is 2

【√（x）+√y）】/【√（x+y）】)2=(x+y+2√xy)\(x+y)

Because x+y 2 (x+y).

So (x+y+2 xy) (x+y) 4 xy 2 xy=2, so the original formula [ (x)+y)] [ (x+y)] 2 so the minimum value is 2

The minimum real number m=27, so that a=b=c=1 3 is obtained, m>=27, and when m=27, we prove, 27(a 3+b 3+c 3)>=6(a 2+b 2+c 2)+1 (1).

3(a^3+b^3+c^3)-(a^2+b^2+c^2)=3(a^3+b^3+c^3)-(a^2+b^2+c^2)(a+b+c)=

a+b)(a-b)^2+(b+c)(b-c)^2+(c+a)(c-a)^2>=0

and 3(a 2 + b 2 + c 2)-1=3(a 2 + b 2 + c 2) - (a + b + c) 2 = (a - b) 2 + (b -c) 2 + (c -a) 2 > = 0

So 9(a 3 + b 3 + c 3) > = 3 (a 2 + b 2 + c 2) > = 1

Proof (1) is sufficient.

Proof: From the fundamental inequality, it can be seen that when x,y 0, x +y 2xy===>2(x²+y²)≥x+y)².

=>√[2(x²+y²)]x+y.Therefore, it can be seen from the inscription, [2(a +b )]a+b.√[2(b²+c²)]b+c,√[2(c²+a²)]c+a.

The addition of the three formulas yields (2)[ a +b )+b +c )+c +a )]2(a+b+c)===>√(a²+b²)+b²+c²)+c²+a²)≥2)(a+b+c).

Do it with the basic properties of inequality.

Under the root a 2+b 2>=(a+b)*(a+b) 2 under the root (a 2+b 2)>=(a+b)*under the root (1 2) under the root (a*a+c*c)>=(a+c)*under the root (1 2) under the root (c*c+b*b)>=(c+b)*under the root (1 2) add the above three formulas.

A 2+B 2 + B 2+C 2+C 2+C 2 + C 2+A 2 under the root number> = 2 * (A + B + C).

f(x) = ln(1+x) -mx

f'(x) = 1/(1+x) -m

f'(1) =0

m = 1/2

f''(x) = - 1/(1+x)^2 < 0 ( for x≠ -1 )

f(x) is defined for x > 1f(x).

Increase (-1, 1 2).

Decrease (1 2,