aligned & f (x)=( x+ i x) ( 3 x+ i 3 x) [ (2 n -1) x+ i (2 n -1) x], & n N Then f ^ (x)=,( Where i= -1 ) aligned

Step-by-step solution

Let f(x) = ( x + i x)( 3x + i 3x) ( (2n-1)x + i (2n-1)x . Using Euler's formula + i = e^ i , we rewrite the product as f(x) = e^ ix e^ i3x e^ i(2n-1)x = e^ ix(1 + 3 + 5 + + (2n-1)) . The sum 1 + 3 + 5 + + (2n-1) is an arithmetic progression of n terms with sum n^2 . Thus, f(x) = e^ in^2x . Differentiating, f'(x) = in^2 e^ in^2x = in^2 f(x) . Taking the second derivative gives f''(x) = in^2 f'(x) = in^2 (in^2 f(x)) = i^2 n^4 f(x) = -n^4 f(x) . Comparing with the options, the correct answer is B .