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Title: Examples for Local Extrema
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 19 | Examples for Local Extrema
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Bright video: https://youtu.be/GkfhOVHU4Rc
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Dark video: https://youtu.be/7k7GE2hyEYg
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Ad-free video: Watch Vimeo video
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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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Python file: Download Python file
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mc19_sub_eng.srt missing
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Timestamps (n/a)
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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) = x_1^2 + 5 x_2^2 $ for all $(x_1, x_2) \in \mathbb{R}^2$. What is the Hessian at the point $(0,0)$?
A1: $$ \begin{pmatrix}2 & 0 \ 0 & 10 \end{pmatrix} $$
A2: $$ \begin{pmatrix}2 & x_1 \ 0 & x_2 \end{pmatrix} $$
A3: $$ \begin{pmatrix}2 & 0 \ 0 & 12 \end{pmatrix} $$
A4: $$ \begin{pmatrix}x_1 & 0 \ 0 & x_2 \end{pmatrix} $$
Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x_1, x_2) = x_1 x_2 $ for all $(x_1, x_2) \in \mathbb{R}^2$. Which statement is correct?
A1: $ \begin{pmatrix}0 & 1 \ 1 & 0 \end{pmatrix} $ is the Hessian matrix at $(0,0)$.
A2: $ \begin{pmatrix}0 & 1 \ 1 & 0 \end{pmatrix} $ is positive definite.
A3: $ \begin{pmatrix}0 & 1 \ 1 & 0 \end{pmatrix} $ is negative definite.
A4: $ f $ has a local extremum at $(0,0)$.
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Last update: 2024-10
