Question

Let g(x) = log(f(x)), where f(x) is twice differentiable function on (0, ∞) such that f(x + 1) = x f(x). Then N = 1, 2, 3... g" (N + 1/2) – g" (1/2) =

(a) –4 {1 + 1/9 + 1/25 + ... + 1/(2N – 1)²}

(b) 4 {1 + 1/9 + 1/25 + ... + 1/(2N – 1)²}

(c) –4{1 + 1/9 + 1/25 + ... + 1/(2N + 1)²}

(d) 4{ 1 + 1/9 + 1/25 + ... +  1/(2N + 1)²}

Answer 1

Correct option (a) –4 {1 + 1/9 + 1/25 + ... + 1/(2N – 1)²}   

Explanation:

We have g(x) = log(f(x))

Now, g(x + 1) – g(x)

= log(f(x + 1)) – log(f(x))