y=4x^2+16/(x^2+1)^2求函数最小值

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y=4x^2+16/(x^2+1)^2求函数最小值
y=4x^2+16/(x^2+1)^2求函数最小值

y=4x^2+16/(x^2+1)^2求函数最小值
a=2x,b=4/(x^2+1)
a^2+b^2>=2ab
当a=b时,a^2+b^2=2ab为最小值
2x=4/(x^2+1)
x^3+x-2=0
(x-1)(x^2+x+2)=0
x=1,当x=1时,
y=4x^2+16/(x^2+1)^2最小值=4+16/4=8

t=x^2>=0
y=2(t+1)+2(t+1)+[4/(t+1)]^2-4>=3[4*(t+1)^2*16/(t+1)^2]^1/3-4=8
2(t+1)=4/(t+1)时取等
即t=2^(1/2)-1

1/16
观察一下等式右边第一第二项都是平方值第二项里也是X^2,X^2>=0，观察16/(x^2+1)^2中，当X^2增加整个16/(x^2+1)^2跟着增加，所以x^2要控制到最小，等于0

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