1.

Find the maximum value of `4sin^2x+3cos^2x+sin(x/2)+cos(x/2)dot`

Answer» `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`
Max`I=x=pi/2`
`I(atx=pi/2)=3+sin^2(pi/2)+sqrt2`
`=3+1+sqrt2`
`=4+sqrt2`.


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