Local Extremums of Real-Valued Functions

Critical Point

Let $f$ be a real-valued function s.t $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$
$x_0$ is a critical point of $f$ if it is not differentiable at $x_0$ or if $Df(x_0)=0$

Local Max

Critical point is a local max if the Hessian is negative-definite:

1. $⟨H_f\cdot v,v⟩<0\mathrm{\forall }v\in {\mathbb{R}}^n\setminus \{0\}$
2. det${(}^{(r)}{H}_f)<0\text{if }r\text{ is odd}$ positive otherwise

Local Min

Critical point is a local min if the Hessian is positive-definite:

1. $⟨H_f\cdot v,v⟩>0\mathrm{\forall }v\in {\mathbb{R}}^n\setminus \{0\}$
2. det${(}^{(r)}{H}_f)>0\mathrm{\forall }r$

Saddle Point

A critical point is a saddle point the Hessian is neither a local max or local min