If `f(x) = cos x cos 2x cos 4x cos 8x cos 16x` then find `f' (pi/4)`

Options

A. `sqrt(2)`
B. `(1)/(sqrt(2))`
C. 1
D. none of these

Answer

A

Explanation

We have,
`f(x)=cosxcos2xcos4xcos8xcos16x`
`implies" "f(x)=cosxcos2xcos2^(2)xcos2^(3)xcos2^(4)x`
`implies" "f(x)=(sin2^(5)x)/(2^(5)sinx)=(sin32x)/(32sinx)`
`implies" "f'(x)=(1)/(32){(32sinxcos32x-cosxsin32x)/((sinx)^(2))}`
`implies" "f'((pi)/(4))=(1)/(32){(32"sin"(pi)/(4)cos4pi-"cos"(pi)/(4)sin8pi)/("sin"^(2)(pi)/(4))}`
`implies" "f'((pi)/(4))=(2)/(sqrt(2))=sqrt(2)`