$ \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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Second Derivative Test

Let $U\subseteq\mathbb{R}^n$ be open and let $f\in C^2(U, \mathbb{R})$. Suppose $\vec{a}\in U$ is a critical point of $f$ and let $Q:\mathbb{R}^n\to\mathbb{R}$ be the quadratic form associated with the Hessian matrix of $f$ at $\vec{a}$. Then
  1. $\vec{a}$ is a local minimum of $f$ if $Q$ is positive definite.
  2. $\vec{a}$ is a local maximum of $f$ if $Q$ is negative definite
  3. $\vec{a}$ is a saddle point of $f$ if $Q$ is indefinite.

Concepts

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Hypothesis

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Results

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Proof

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Second Derivative Test

Let $U\subseteq\mathbb{R}^n$ be open and let $f\in C^2(U, \mathbb{R})$. Suppose $\vec{a}\in U$ is a critical point of $f$ and let $Q:\mathbb{R}^n\to\mathbb{R}$ be the quadratic form associated with the Hessian matrix of $f$ at $\vec{a}$. Then
  1. $\vec{a}$ is a local minimum of $f$ if $Q$ is positive definite.
  2. $\vec{a}$ is a local maximum of $f$ if $Q$ is negative definite
  3. $\vec{a}$ is a saddle point of $f$ if $Q$ is indefinite.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Second Derivative Test

Let $U\subseteq\mathbb{R}^n$ be open and let $f\in C^2(U, \mathbb{R})$. Suppose $\vec{a}\in U$ is a critical point of $f$ and let $Q:\mathbb{R}^n\to\mathbb{R}$ be the quadratic form associated with the Hessian matrix of $f$ at $\vec{a}$. Then
  1. $\vec{a}$ is a local minimum of $f$ if $Q$ is positive definite.
  2. $\vec{a}$ is a local maximum of $f$ if $Q$ is negative definite
  3. $\vec{a}$ is a saddle point of $f$ if $Q$ is indefinite.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon

Second Derivative Test

Let $U\subseteq\mathbb{R}^n$ be open and let $f\in C^2(U, \mathbb{R})$. Suppose $\vec{a}\in U$ is a critical point of $f$ and let $Q:\mathbb{R}^n\to\mathbb{R}$ be the quadratic form associated with the Hessian matrix of $f$ at $\vec{a}$. Then
  1. $\vec{a}$ is a local minimum of $f$ if $Q$ is positive definite.
  2. $\vec{a}$ is a local maximum of $f$ if $Q$ is negative definite
  3. $\vec{a}$ is a saddle point of $f$ if $Q$ is indefinite.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof