Question about Fermi energy and heat capacity of a Fermi gas

[Clara Chung]

Homework Statement

Attached below, I need help on c and d,
I am not sure about the condition dN/dT=0 in part d. I mean.. of course dN/dT=0 if N is fixed... but I am not sure that the fermi distribution I am using represent a condition with fix N... Moreover, isn't N allow to change because we are imposing the condition that only V is constant when calculating the heat capacity?

Relevant Equations

Attached

Attempt:

 

[stevendaryl]

This was posted awhile ago. Did your instructor give you the answer? I have thought about it, and I really don't understand how this result is derived. The problem with computing $\frac{df}{dT}$ is that $f$ depends on $\mu$, which itself may change with temperature.

 

[Clara Chung]

This was posted awhile ago. Did your instructor give you the answer? I have thought about it, and I really don't understand how this result is derived. The problem with computing $\frac{df}{dT}$ is that $f$ depends on $\mu$, which itself may change with temperature.

He didn't give me the answer. However he said my approach is right and the reason why we can handle N as if it were a deterministic variable is that we are always interested in the thermodynamic limit, where all variables, including N, take a well-defined value. I think the question also assume u is a constant...