函数f（x）=2/1（sinxcosx）+2/1丨sinx-cosx丨,则f（x）的值域为?

Question

函数f（x）=2/1（sinxcosx）+2/1丨sinx-cosx丨,则f（x）的值域为?

Answer 1

f（x）=1/2*（sinxcosx）+1/2丨sinx-cosx丨
设|sinx-cosx|=t
那么(sinx)^2-2sinxcosx+(cosx)^2=t^2
∴1-2sinxcosx=t^2
那么sinxcosx=(t^2-1)/2
又t=|sinx-cosx|
=√2|√2/2sinx-√2/2cosx|
=√2|sin(x-π/4)|
∴t∈[0,√2]
∴y=1/4(t^2-1)+1/2t
=1/4t^2+1/2t-1/4
=1/4(t+1)^2-1/2
∵t∈[0,√2]
∴t=0时,ymin=-1/4
t=√2时,ymax=1/4+√2/2
即函数值域为[-1/4,1/4+√2/2]