函数y=4/cos^2+9/sinx^2的最小值

题目

函数y=4/cos^2+9/sinx^2的最小值
y=4/cosx^2+9/sinx^2

答案

y=4/cos^2+9/sinx^2
=4(cos^2x+sin^2x)/cos^2x+9(cos^2x+sin^2x)/sinx^2
=4+4tan^2x+9+9/tan^2x
≥13+2√36
=25
当且仅当4tan^2x=9/tan^2x时成立,即tan^2x=3/2