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Problems related to Maxwell relations

1. Homework Statement
Given the entropy of a system :
$$ S = AU^αV^βN^{1-α-β} $$
The problem requires me to write
$$ (\frac{∂T}{∂U})_{V,N} > 0,  (\frac{∂P}{∂V})_{U,N} < 0, (\frac{∂μ}{∂N})_{U,V} > 0$$
to find the mathematical constraint of α and β
2. Homework Equations
dU = TdS - PdV + μdN

3. The Attempt at a Solution
Actually, I don't quite understand the rationale behind Maxwell relations.
I tried to write
dF = SdT - PdV + μdN
I can then get
$$ (\frac {∂T} {∂F})_{V,N}$$
but I just don't know how to transform the variable. Can anyone give me an idea of these kind of problems. Thanks!
 
Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 
This is actually a bonus question in my HW. It wrote something but I think they are wrong. The solution should be ready in the coming few days. I will share the ideas once it is released.
 
Thanks! I will try it!
 

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