Problems related to Maxwell relations

[Mayan Fung]

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

Homework Equations

dU = TdS - PdV + μdN

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!

 

[Mayan Fung]

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.

 

[HenryWhu]

you can check it.http://odysriwo8.bkt.clouddn.com/wo.PNG

 

[Mayan Fung]

Thanks! I will try it!