First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the real numbers, let $w:U\to\mathbb{R}$ be of class $C^2$ on $U$, and let $x\in U$. Regard $\mathbb{R}^n$ as a metric space through the Euclidean distance $d_E$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$.

Let $0_{\mathbb{R}^n}$ denote the origin of $\mathbb{R}^n$, that is, the point all of whose coordinates are $0$, and let $0_n$ denote the real $n\times n$ matrix all of whose entries are $0$. Since its entries satisfy $(0_n)_{ij}=(0_n)_{ji}$, the matrix $0_n$ is symmetric, and the Hessian matrix $D^2w(x)$ is symmetric by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian; both therefore lie in the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, so the positive semidefinite ordering $\preceq$ applies to them.

Then the following hold, with $Dw(x)$ the gradient of $w$ at $x$.

1. (Local maximum) If $w$ has a local maximum at $x$ relative to $U$, then

$$Dw(x)=0_{\mathbb{R}^n}\qquad\text{and}\qquad D^2w(x)\preceq 0_n .$$

2. (Local minimum) If $w$ has a local minimum at $x$ relative to $U$, then

$$Dw(x)=0_{\mathbb{R}^n}\qquad\text{and}\qquad 0_n\preceq D^2w(x).$$