The function `f(x) = sin ^(4)x+ cos ^(4)x ` increases, if

Options

A. `0 lt x lt (pi)/8`
B. `(pi)/4lt x lt (3pi)/8`
C. `(3pi)/8ltx lt (5pi)/8`
D. `(5pi)/8 lt x lt (3pi)/4`

Answer

B

Explanation

Given `f(x)=sin^(4)x+cos^(4)x`
`=(sin^(2)x+cos^(2)x)^(2)-2sin^(2)x cos^(2)x`
`=1-(sin^(2)2x)/2=1+((1-cos4x))/4=(3+cos4x)/4`
`:.f'(x)=-sin4x`
If `f(x)` increases then `f'(x) gt0`
`implies-sin4xgt0`
`impliessin4xlt0`
`implies (2n+1)pilt4xlt(2n+2)pi`
`implies ((2n+1)pi)/4 lt x lt ((n+1)pi)/2`
FOr `n=0, (pi)/4 lt x lt (pi)/2`
`:.f(x)` is increasing in `((pi)/4, (3pi)/8) [ :' (pi)/2=(4pi)/8 gt (3pi)/8]`