Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that, if $\varphi_\alpha''(0) > 0$, then there exists $\eta > 0$ such that for all $t \in ]-\eta, \eta[$,
$$\frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM) \geq \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A) - \operatorname{Tr}(A^{-1}M) \operatorname{det}^{-\alpha}(A) t.$$