> [!NOTE] Definition (Local Maximum of Surface in $\mathbb{R}^{3}$) > Let $f:U\subset \mathbb{R}^{2}\to \mathbb{R}$ be a [[Real-Valued Function on Real n-Space (Multivariable Function)|multivariable function]]. Let $\underline{c}\in \mathbb{R}^{2}$ be a [[Critical Point of Real-Valued Function on Real 2-Space|critical point]] of $f.$ Then $\underline{c}$ is a local maximum iff there exists $\delta>0$, such that for all $\underline{x}\in U$ $ ||\underline{x}-\underline{c}| | <\delta \implies f(x) \leq f(c)$where $||\dots||$ denotes [[Euclidean Norm|length]].