证明不等式：(x²+4/y²)(y²+1/x²)>=9

证明不等式：(x²+4/y²)(y²+1/x²)>=9
证明不等式：(x²+4/y²)(y²+1/x²)>=9

证明不等式：(x²+4/y²)(y²+1/x²)>=9
(x²+4/y²)(y²+1/x²)>=9
证明
(x²+4/y²)(y²+1/x²)
=x²y²+4+1+4/(x²y²)
=x²y²+4/(x²y²)+5
∵x²y²>0
∴x²y²+4/(x²y²)≥2√4=4
x²y²=2时取等
∴原式
x²y²+4/(x²y²)+5
≥4+5
=9

依Cauchy不等式得
(x²+4/y²)(1/x²+y²)≥[x·(1/x)+(2/y)·y]²
∴(x²+4/y²)(1/x²+y²)≥9.