I am currently working with a class of functions, where every function looks like
$f(x)=v^T(xA+(1-x)B)^{-1}v,$
where $v\in\mathbb{R}^n$ is an arbitrary vector, $A,B \in \mathbb{R}^{n\times n}$ are positive definite matrices and $x\in(0,1)$. Is there a straight forward way to prove, that $f(x)$ is convex on $(0,1)$?