# Proof that the heat capacity of a system at constant volume is greater than zero at the equilibrium

I'm trying to prove that the heat capacity at constant volume, given by

\begin{equation} C_{_V} = \left(\frac{\partial U}{\partial T}\right)_{V}, \end{equation}

where $U$ is the energy of the system and $T$ it's temperature, is greater than zero when the system is at equilibrium.

To do so I started considering

\begin{equation} \left(\frac{\partial^{2} S}{\partial U^{2}}\right)_{_V} = -\frac{1}{T^{2}}\left(\frac{\partial T}{\partial U}\right)_{_V} = -\frac{1}{C_{_V}T^{2}} \end{equation}

and so, I have to prove that

\begin{equation} \left(\frac{\partial^{2} S}{\partial U^{2}}\right)_{_V} <0 \end{equation}

at equilibrium. Now, at equilibrium, the entropy is maximum for any given value of the energy $U$, which can be stated as

\begin{equation} \left(\frac{\partial S}{\partial X}\right)_{_U} = 0 \,\,\,\,\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,\,\,\,\, \left(\frac{\partial^{2} S}{\partial X^{2}}\right)_{_U}<0 \end{equation}

where $X$ is any extensive parameter of the system. This means that, for a given value of $U$, at equilibrium, the entropy is maximum with respect to this paramenter. Since $U$ is also an extensive paramenter of the system can I write the maximum entropy principle like

\begin{equation} \left(\frac{\partial^{2} S}{\partial U^{2}}\right)_{_X}<0 \end{equation} ?

Thank you.