已知函数f(x)=(2sin2x+根号3)cosx-sin3x,求f(x)的最值.2)若f(x)=根号3,x属于(0,派)求x
问题描述:
已知函数f(x)=(2sin2x+根号3)cosx-sin3x,求f(x)的最值.2)若f(x)=根号3,x属于(0,派)求x
答
f(x)=2(sin2x+√3/2)cosx-sin3x
=2sin2xcosx+√3cosx-sin3x
=sin3x+sinx+√3cosx-sin3x
=2sin(x+π/3)
x的最小值为2.最大值为-2
∴x+π/3=2kπ+π/2,k∈Z,
x=2kπ+π/6,k∈Z,f(x)max=2
x+π/3=2kπ-π/2,
x=2kπ-5π/6,k∈Z,f(x)min=-2
f(x)=根号3
第二问.2sin(x+π/3) = 根号3
sin(x+π/3) =根号3/2
sin(π/3)=sin(2π/3)=根号3/2
因为x属于(0,派)
所以x=π/3