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Title: Local Extrema
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Series: Multivariable Calculus
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Chapter: Extrema of Functions
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YouTube-Title: Multivariable Calculus 18 | Local Extrema
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Bright video: Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mc18_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Definitions in the video: local maximum, local minimum, isolated local minimum, isolated local maximum, local extremum, critical point, positive definite matrix, indefinite matrix
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Timestamps
00:00 Introduction
00:42 Explaining local extrema
01:51 Definition: Local maximum and local minimum
03:10 Correction: Isolated and strict local maxima should be distinguished, see PDF!
04:12 Definition: Local extremum
04:44 Necessary condition for local extremum
06:12 Sufficient condition for local extremum
08:31 Positive definite Hessian
09:30 Negative definite Hessian
09:54 Indefinite definite Hessian
11:13 Implication from having a local extremum
12:59 Outro
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x_1, x_2) = 4$ for all $(x_1, x_2) \in \mathbb{R}^2$. Does $f$ has a local maximum at the point $(1,3)$?
A1: Yes!
A2: No, but it’s an isolated local minimum.
A3: No, the function has no local maxima.
A4: One needs more information.
Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x_1, x_2) = x_1^2 + x_2^2 $ for all $(x_1, x_2) \in \mathbb{R}^2$. Does $f$ has a local maximum at the point $(0,0)$?
A1: Yes!
A2: No, but it’s an isolated local minimum.
A3: No, the function has no local maxima.
A4: One needs more information.
Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = \cos(x_1 x_2)$. What is the Hessian at $(0,0)$?
A1: $ \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$
A2: $ \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix}$
A3: $ \begin{pmatrix} -1 & 0 \ 0 & 2 \end{pmatrix}$
A4: $ \begin{pmatrix} 1 & 2 \ 0 & 2 \end{pmatrix}$
A5: $ \begin{pmatrix} 1 & 0 \ 0 & 2 \end{pmatrix}$
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Last update: 2026-04
