x属于0到π\2的闭区间，求y=4根号2sinxcosx+cos2x的值域

y=4根号2sinxcosx+cos2x
=2根号2sin2x+cos2x
=3(2根号2/3*sin2x+1/3*cos2x)
=3sin(2x+m)
式中sinm=1/3
0<m<30°

0<=x<=π/2
m<=2x+m<=π+m
sin(π+m)<=sin(2x+m)<=1
sin(π+m)=-sinm=-1/3
-1/3<=sin(2x+m)<=1
-1<=y<=3.

值域是[-1,3].

y=4根号2sinxcosx+cos2x
=4根号(sin2x+cos2x)
=4√[√2sin(2x+π/4)]
y=4根号2sinxcosx+cos2x的值域:[0,4√√2]