設(shè)y=2sinα(cosα)^2,
則y^2=4 (sinα)^2(cosα)^4,
y^2=2*2(sinα)^2*(cosα)^2*(cosα)^2
利用三元基本不等式可得下式
≤2*[(2(sinα)^2+(cosα)^2+(cosα)^2)/3]^3
=2*(2/3)^3=16/27,
∴-4√3/9≤y≤4√3/9.
即-4√3/9≤2sinα(cosα)^2≤4√3/9.求導(dǎo)的話可以做出來(lái)嗎？令f(x)= 2sinx * ( cosx)^2= 2sinx *[1-(sinx)^2]=2sinx-2(sinx)^3令sinx=t(t∈[-1,1])f(x)=2t-2t^3,f'(x)=2-6t^2當(dāng)t∈[-1,-1/√3)時(shí)，f'(x)＜0當(dāng)t∈(-1/√3,1/√3)時(shí)，f'(x)＞0當(dāng)t∈(1/√3,1]時(shí),f'(x)＜0所以當(dāng)-1<=t<-1/根號(hào)3或1>=t>1/根號(hào)3時(shí),函數(shù)單調(diào)遞減；當(dāng)-1/根號(hào)3< t<1/根號(hào)3，函數(shù)單調(diào)遞增?！鄁(x)的極大值為f(1/√3)，此時(shí)最大值為f(-1)或f(1/√3),f(-1)=0,f(1/√3)=4√3/9最大值應(yīng)為4√3/9.f(x)的極小值為f(-1/√3)，此時(shí)最小值為f(1)或f(-1/√3),f(1)=0,f(-1/√3)=-4√3/9最小值應(yīng)為-4√3/9。