I have a real positive quantity $f(A)= tr[A^{-1} (\partial_x A) A^{-1} (\partial_x A)] + 2v^T A^{-1} v$ where $x$ is a real parameter and $A$ is a symmetric matrix. Let us assume that $S$ is a shift matrix (possibly a function of x) such that $A + S$ is symmetric
$$f(A + S)= tr[(A + S)^{-1} \partial_x (A + S) (A + S)^{-1} \partial_x (A + S)] + 2v^T (A + S)^{-1} v.$$
How can one prove or disprove that $f(A) \le f(A+S)$?