2 Limits/Differenciability questions

[hellking4u]

1. The problem statement, all variables and given/known data
Q1. let g(x) = log(f(x)), where f(x) is a twice diffrenciable positive function on (0, inf) such that f(1+x) = xf(x)

Then for N = 1,2,3.....

g''(N+1/2) - g''(1/2) = ??

Q2. Let f(x) be differenciable on the interval (0, inf) such that f(1) = 1, and Lim(t-->x) [t^2f(x)-x^2f(t)]/t-x = 1 for each x > 0

Then f(x) is...??


2. Relevant equations
none I believe....?


3. The attempt at a solution

I tried Q1 by finding g''(x) and f''(x) and then putting them into the raw equation, using the given condition f(1+x) = xf(x) and writing
g''(1/2) as g''(-1/2+1)
and g''(N+1/2) as (n-1/2 +1)

But to no avail


P.S. The problems are from a MCQ test......tell me if you'd need the options aswell....I'll be happy to provide them! :)

[LCKurtz]

Q2. Let f(x) be differenciable on the interval (0, inf) such that f(1) = 1, and Lim(t-->x) [t^2f(x)-x^2f(t)]/t-x = 1 for each x > 0

Then f(x) is...??




I assume you mean with parentheses (t-x) in the denominator.
Try adding and subtracting x^2f(x) in that numerator, should lead you to a first order linear DE.