$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Let $A\subseteq\mathbb{R}^n$, $f:A\to\mathbb{R}^m$, and $\vec{a}\in \text{int}(A)$. Then $\vec{a}$ is a local maximum if there exists some $delta>0$ such that $f(\vec{x})\leq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$, and $\vec{a}$ is a local minumum if there exists $\delta>0$ such that $f(\vec{x})\geq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$. Either of the two cases are called local extremum.

Concepts

Coming soon

Used In

Coming soon

Hypothesis

  • If $f(\vec{a})$ is a local maximum or minimum, then $\vec{a}$ is a critical point.

Results

  • If the associated Hessian matrix is positive definite, then local minimum.
Let $A\subseteq\mathbb{R}^n$, $f:A\to\mathbb{R}^m$, and $\vec{a}\in \text{int}(A)$. Then $\vec{a}$ is a local maximum if there exists some $delta>0$ such that $f(\vec{x})\leq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$, and $\vec{a}$ is a local minumum if there exists $\delta>0$ such that $f(\vec{x})\geq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$. Either of the two cases are called local extremum.

Concepts

Coming soon

Used In

Coming soon

Hypothesis

  • If $f(\vec{a})$ is a local maximum or minimum, then $\vec{a}$ is a critical point.

Results

  • If the associated Hessian matrix is positive definite, then local minimum.
FullPage
definition
concepts
used in
hypothesis
results
FullPage
definition
concepts
used in
hypothesis
results
Let $A\subseteq\mathbb{R}^n$, $f:A\to\mathbb{R}^m$, and $\vec{a}\in \text{int}(A)$. Then $\vec{a}$ is a local maximum if there exists some $delta>0$ such that $f(\vec{x})\leq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$, and $\vec{a}$ is a local minumum if there exists $\delta>0$ such that $f(\vec{x})\geq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$. Either of the two cases are called local extremum.

Concepts

Coming soon

Used In

Coming soon

Hypothesis

  • If $f(\vec{a})$ is a local maximum or minimum, then $\vec{a}$ is a critical point.

Results

  • If the associated Hessian matrix is positive definite, then local minimum.
Let $A\subseteq\mathbb{R}^n$, $f:A\to\mathbb{R}^m$, and $\vec{a}\in \text{int}(A)$. Then $\vec{a}$ is a local maximum if there exists some $delta>0$ such that $f(\vec{x})\leq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$, and $\vec{a}$ is a local minumum if there exists $\delta>0$ such that $f(\vec{x})\geq f(\vec{a})$ for all $\vec{x}\in B_{\delta}(\vec{a})$. Either of the two cases are called local extremum.

Concepts

Coming soon

Used In

Coming soon

Hypothesis

  • If $f(\vec{a})$ is a local maximum or minimum, then $\vec{a}$ is a critical point.

Results

  • If the associated Hessian matrix is positive definite, then local minimum.
FullPage
definition
concepts
used in
hypothesis
results