已知函数fx=2sinx/2cosx/2+cosx,其中x属于R求fx最小正周期和值域.

Question

已知函数fx=2sinx/2cosx/2+cosx, 其中x属于R 求fx最小正周期和值域.

Answer 1

f(x)=2sin(x/2)cos(x/2)+cosx
=sinx+cosx
=√2[(√2/2)sinx+(√2/2)cosx]
=√2(sinxcosπ/4+cosxsinπ/4)
=√2sin(x+π/4)
最小正周期=2π/1=2π
sin(x+π/4)=1时,f(x)有最大值f(x)max=√2
sin(x+π/4)=-1时,f(x)有最小值f(x)min=-√2
函数的值域为[-√2,√2]