若π/4〈X〈π/2,则函数Y=tan2X*tan³X的最大值为什么

问题描述:

若π/4〈X〈π/2,则函数Y=tan2X*tan³X的最大值为什么
tan2x和tanx的立方中间是乘号

Y=tan2X*tan³X
=2tan^4 x/(1-tan^2 x)
令t=tan^2 x>1,
y=2t^4/(1-t^2)
=[2(t^2+1)(t^2-1)+2]/(1-t^2)
=-2(t^2+1)+2/(1-t^2)
=-2[(t^2-1)+1/(t^2-1)+2]
≤-2[2+2]
=-8
当(t^2-1)=1/(t^2-1),t=√2时,y max=-8