"Let f : S $\to$ S where S = (0, $\infty$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $\to$ R be defined as g(x) = loge f(x), then the value of |g''(5) $-$ g''(1)| is equal to :"

Question

Accepted Answer

"$f(x + 1) = xf(x)$

$\ln (f(x + 1)) = \ln x + \ln f(x)$

$g(x + 1) = \ln x + g(x)$

$g(x + 1) - g(x) = \ln x$ ..... (i)

$g'(x + 1) - g'(x) = {1 \over x}$

$g''(x + 1) - g''(x) = {{ - 1} \over {{x^2}}}$

$g''(2) - g'(1) = {{ - 1} \over 1}$ .... (ii)

$g''(3) - g''(2) = {{ - 1} \over 4}$ .... (iii)

$g''(4) - g''(3) = {{ - 1} \over 9}$ ..... (iv)

$g''(5) - g''(4) = {{ - 1} \over {16}}$ ....(v)

Adding (ii), (iii), (iv) & (v)

$$g''(5) - g''(1) = - \left( {{1 \over 1} + {1 \over 4} + {1 \over 9} + {1 \over {16}}} \right) = {{ - 205} \over {144}}$$

$|g''(5) - g''(1)|\, = {{205} \over {144}}$"

Let f : S $\to$ S where S = (0, $\infty$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $\to$ R be defined as g(x) = log_e f(x), then the value of |g''(5) $-$ g''(1)| is equal to :

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Let f : S $\to$ S where S = (0, $\infty$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $\to$ R be defined as g(x) = loge f(x), then the value of |g''(5) $-$ g''(1)| is equal to :

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Let f : S $\to$ S where S = (0, $\infty$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $\to$ R be defined as g(x) = log_e f(x), then the value of |g''(5) $-$ g''(1)| is equal to :