The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x$, then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x$, then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.
The Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables.
Specifically, suppose $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is a function taking as input a vector $x \in \mathbb{R}^n$ and outputting a scalar $f(x) \in \mathbb{R}$; if all second partial derivatives of $f$ exist and are continuous over the domain of the function, then the Hessian matrix $H$ of $f$ is a square $n \times n$ matrix
