Let $U$ be an open convex subset of $\mathbb{R}^n$, and $f: U\rightarrow\mathbb{R}$ be a function having continuous first and second partial derivatives on $U$. Let $H_{f(X)}$ denote the Hessian of $f$ at $X\in U$. If $\det(H_{f(X)})\neq 0$ for all $X\in U$, can we conclude that $v^tH_{f(X)}v\neq0$ for all $X\in U$ and $v\neq 0$, $v\in\mathbb{R}^n$.
I found this in the discussion of absolute maxima or absolute minima of functions on convex open sets (page number 43), in the book `Multivariate Calculus and Geometry(Third Edition)' by Sean Dineen.