Extremum

Definition

Maximum

Definition

Consider an Partially Ordered Set $X$ and a subset $Y\subset X$. The largest element (maximum) $y_0$ of $Y$ is defined as $y_0\text{s.t.}{y}_0\in Y\text{and}\forall y\in Y,y\le y_0$

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Minimum

Definition

Consider an Partially Ordered Set $X$ and a subset $Y\subset X$. The smallest element (minimum) $y_0$ of $Y$ is defined as $y_0\text{s.t.}{y}_0\in Y\text{and}\forall y\in Y,y\ge y_0$

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Local Maximum

Definition

Suppose $f:\mathcal{D}\to \mathbb{R}$. The function $f$ has local maximum at $x_0$ if $\exists ϵ>0,(\forall x\in \mathcal{D},|x_0-x|<ϵ)\Rightarrow f(x)\le f(x_0)$

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Local Minimum

Definition

Suppose $f:\mathcal{D}\to \mathbb{R}$. The function $f$ has local minimum at $x_0$ if $\exists ϵ>0,(\forall x\in \mathcal{D},|x_0-x|<ϵ)\Rightarrow f(x_0)\le f(x)$

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