解:令y=(sinα)^2cosα,得:2y^2=2(sinα)^4(cosα)^2=2(sinα)^2(sinα)^2(cosα)^2≤{[2(cosα)^2(sinα)^2(sinα)^2]/3}^3=(2/3)^3=8/27(三次基本不等式)2y^2≤8/27y^2≤4/27ymax=2√3/9,当且仅当2(cosα)^2=(sinα)^2,即tanα=√2,α=arctan√2时取到最大值
