Question

f（x）=sin^3+3sinx的值域是

f（x）=sin^3 x+3cosx 不好意思·~~~发错啊

Answer 1

f（x）=sin^3x+3cosx
f'(x)=3sin²xcosx-3sinx=3sinx(sinxcosx-1)
∵sinxcosx=1/2*sin2x≤1
∴sinxcosx-1<0
令f'(x)=0,即sinx=0===>x=kπ,k∈Z
∴x∈(2kπ,(2k+1)π),f'(x)<0,f(x)递减
x∈（(2k+1)π，2kπ,+2π),f'(x)>0,f(x)递增
∴x=±π,±3π,±5π,....时，f(x)min=-3
x=0,±2π,±4π,..........时，f(x)max=3
∴函数值域为[-3,3]