The maximum value of `4sin^2x+3cos^2x+sin(x/2)+cos(x/2)` is

Explanation

`I=4sin^2x+3(1-sin^2x)+sin(x/2)+cos(x/2)`

`I=4sin^2x+3-3sin^2x+sin(x/2)+cos(x/2)`

`I=sin^2x+3+sin(x/2)+cos(x/2)`

`I=sin^2x+3+sqrt2[1/sqrt2sin(x/2)+cos(x/2)*1/sqrt2]`

`=3+sin^2x+sqrt2sin[pi/4+x/2]`

`sin(pi/4+x/2)=1`

When `x=pi/2`

i.e.`sin(pi/4+pi/4)=sin(pi/2)=1`

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