entropy maximum postulate is a statement about the behaviour of the entropy function along off-equilibrium curves in $\mathcal{M}$, not along equilibrium sequences $\mathcal{M}_{eq}$. Therefore we need an expression for an extended Massieu function $\hat{Z} = \hat{Z}(X^\rho, \mu^i)$ where $X^\rho$ denote a set of off-equilibrium variables. The equillibrium configurations occur at the values $X^\rho_{eq} = X^\rho_{eq}(\mu^i)$ which are solutions of $$\frac{\partial \hat{Z}}{\partial X^\rho} = 0.$$ A stable equilibrium takes place at the point $X^\rho_{eq}$ if and only if it is a local maximum of $\hat{Z}$ at fixed $\mu^i$. Therefore stable solutions have a matrix $$\left. \frac{\partial^2 \hat{Z}(X^\tau, \mu^i)}{\partial X^\rho \partial X^\sigma}\right|_{X^\tau = X^\tau_{eq}(\mu^i)}
 \label{eq:matrix}$$ with a negative spectrum of eigenvalues. A change of stability takes place when one of these eigenvalues become zero and changes sign. It has been shown that we can obtain information about the sign of an eigenvalue near a point where it vanishes without computing the spectrum of the matrix (\[eq:matrix\]). What we have to do is to plot a conjugate variable $\beta_a (\mu^a) = \frac{\partial Z_{eq}}{\partial \mu^a}$ against a control parameter $\mu^a$ along an equilibrium sequence for some fixed $a$. When a change of stability occurs, called a turning point, the plot of the stability curve 