Let f(x) be a differentiable function such that
`f'(x)=sinx+sin4xcosx.` Then `f'(2x^(2)+(pi)/(2))"at "x=sqrt((pi)/(2))` is equal to

Options

A. 0
B. -1
C. `-2sqrt(pi)`
D. none of these

Answer

C

Explanation

We have,
`f'(x)=sinx+sin4xcosx" for all "x.`
`implies" "(d)/(dx)(f(x))dx=sinx+sin4xcosx" for all "x" "...(i)`
Now,
`f'(2x^(2)+(pi)/(2))=(d)/(dx){f(2x^(2)+(pi)/(2))}`
`=(d)/(d(2x^(2)+(pi)/(2))){f(2x^(2)+(pi)/(2))}.(d)/(dx)(2x^(2)+(pi)/(2))`
`={sin(2x^(2)+(pi)/(2))+sin(8x^(2)+2pi)cos(2x^(2)+(pi)/(2))}xx4x`
`" "["Using (i)"]`
`-4x{cos2x^(2)-sin8x^(2)sin2x^(2)}`
`{f'(2x^(2)+(pi)/(2))}_("at "x=sqrt((pi)/(2)))=4sqrt((pi)/(2)){c ospi-sin4pisinpi}`
`=-2sqrt(2pi)`