Let $f$ be a function on positive definite cone (a set of $p\times p$ positive definite symmetric matrix), which is defined by $f(A)=\log |A|+\text{tr}(SA^{-1})$, where $S$ is some fixed positive definite symmetric matrix. I know this function has a global minimum and the minimizer can be easily found by solving $\text{grad}f=0$. But what makes this method reasonable? I mean even if the gradient is 0, this does not ensure that $f$ has a minimizer at such a critical point. Also the given function $f$ is not convex. In case $p=1$, it can easily verified that $f$ has a minimizer at the desired critical point.
For general $p$, my attempt was that since we know that a critical point of $f(A)=\log |A|+\text{tr}(SA^{-1})$ is unique, I've tried to show Hessian matrix at such a critical point is positive definite, which would imply that the critical point is an unique minimizer. I've tried this by considering $A$ as a $\frac{p(p+1)}{2}$-vector, but this was not easy due to the term $A^{-1}$. Can anyone give me a hint with my attempt or any new idea? Thanks in advance.