求证明 x²y²≤(x²+y²)/4

题目

求证明 x²y²≤(x²+y²)/4

答案

如果有x>0,y>0,则(x-y)²≥0↔x²+y²≥2xy↔x²+2xy+y²≥4xy↔xy≤(x+y)²/4.将x换成x²、y换成y²,得∴x²y²≤(x²+y²)²/4....