f(x)=2sinx(sinx+cosx),x∈[-π/2,π/2]时,求f(x)的值域

题目

f(x)=2sinx(sinx+cosx),x∈[-π/2,π/2]时,求f(x)的值域

答案

f(x)=2sin^2x+2sinxcosx
=1-cos2x+sin2x
=√2sin(2x-45°）+1
x∈[-π/2,π/2]
2x∈[-π,π]
2x-45∈[-5π/4,3π/4]
sin(2x-45)∈[-1,1]
f(x)∈[1-√2,1+√2]