Let $A \in S_n^{++}(\mathbf{R})$ and let $f : t \mapsto \ln(\operatorname{det}(I_n + tA))$. Show that $$\forall t \in \mathbf{R}_+, \quad \ln(\operatorname{det}(I_n + tA)) \leq \operatorname{Tr}(A) t.$$
Let $A \in S_n^{++}(\mathbf{R})$ and let $f : t \mapsto \ln(\operatorname{det}(I_n + tA))$. Show that
$$\forall t \in \mathbf{R}_+, \quad \ln(\operatorname{det}(I_n + tA)) \leq \operatorname{Tr}(A) t.$$