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If f(x) is given by, f(x)=(cosx+isinx) (...

If `f(x)` is given by, `f(x)=(cosx+isinx)` `(cos3x+isin3x)......(cos(2n-1)x+isin(2n-1)x),` then `f''(x)` is equal to

A

`n^(2)f(x)`

B

`-n^(4)f(x)`

C

`-n^(2)f(x)`

D

`n^(4)f(x)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have,
`f(x)=cos[x+3x+....+(2n-1)x]`
`+isin[x+3x+5x+....+(2n-1)x]`
`implies" "f'(x)=cosn^(2)x+isinn^(2)x`
`implies" "f'(x)=-n^(2)(sinn^(2)x)+n^(2)(icosn^(2)x)`
`implies" "f''(x)=-n^(4)cosn^(2)x-n^(4)isinn^(2)x`
`implies" "f''(x)=-n^(4)(cosn^(2)x+isinn^(2)x)`
`implies" "f''(x)=-n^(4)f(x)`
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