2 Limits/Differenciability questions

[hellking4u]

Homework Statement

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...??

Homework Equations

none I believe...?

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.

 

[Architeuthis Dux]

I would approach these questions by first understanding the definitions and concepts involved. In Q1, we are given a function g(x) defined as the logarithm of another function f(x), which is twice differentiable and positive on the interval (0, inf). We are also given the condition that f(1+x) = xf(x). This condition can be rewritten as f(x) = (x-1)f(x-1), which implies that f(x) is a multiple of the function (x-1). This is important because it tells us that f(x) is not just any arbitrary function, but it has a specific form.

Using this information, we can find the derivatives of g(x) and f(x) and substitute them into the given equation. We can then use the product rule and chain rule to simplify the equation and solve for g''(N+1/2) - g''(1/2). This would give us an expression in terms of f(x) and its derivatives.

In Q2, we are given a function f(x) that is differentiable on the interval (0, inf) and satisfies a given limit condition. We are asked to determine what type of function f(x) is. To do this, we can use the definition of the limit and the properties of differentiability to simplify the given equation. This would involve using the product rule, chain rule, and the definition of differentiability. Once we have simplified the equation, we can compare it to the definition of a specific type of function to determine what type of function f(x) is.

In both cases, it is important to understand the concepts involved and use the given information to simplify the equations and find the solutions. It may also be helpful to sketch the functions and visualize the given conditions to gain a better understanding of the problem.