\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - 3-Space Extrema

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Extrema In 3-Space

For a full lecture on this topic, including optimization, here is a great video.

Prof Leonard - Finding Extrema of Functions of 2 Variables (Max and Min)

video by Prof Leonard

In order to determine extrema of functions in 3-space, we do something similar to the first and second derivative tests that you used calculus 1. Let's start with the first derivative. Here is a video that explains it very well.

Thomas Wernau - Notes on Relative Extrema Part 1 Finding Critical Points

video by Thomas Wernau

For the second derivative, the equations are a bit different than you are used to but the idea parallels the single variable version. Here is a video explaining it.

Thomas Wernau - Notes on Relative Extrema Part 2 Second Derivative Test

video by Thomas Wernau

Here is a proof of the second derivative test. You do not need to know this in order to use the second derivative. However, it will help you better understand the test. So we recommend that you watch it.

Thomas Wernau - Notes on Relative Extrema Part 2 Second Derivative Test

video by Thomas Wernau

Math Word Problems Demystified

Practice

Find the critical points of \(f(x,y)=xy\)

Problem Statement

Find the critical points of \(f(x,y)=xy\)

Solution

Thomas Wernau - 4360 video solution

video by Thomas Wernau

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Classify the critical points of \(f(x,y) = xy\).

Problem Statement

Classify the critical points of \(f(x,y) = xy\).

Solution

Thomas Wernau - 4363 video solution

video by Thomas Wernau

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Find the critical points of \(f(x,y) = 4+x^3+y^3-3xy\)

Problem Statement

Find the critical points of \(f(x,y) = 4+x^3+y^3-3xy\)

Solution

Thomas Wernau - 4361 video solution

video by Thomas Wernau

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Classify the critical points of \(f(x,y) = 4+x^3+y^3-3xy\)

Problem Statement

Classify the critical points of \(f(x,y) = 4+x^3+y^3-3xy\)

Solution

Thomas Wernau - 4364 video solution

video by Thomas Wernau

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Classify the critical points of \(f(x,y) = 1-(x^2+y^2)^{1/3}\)

Problem Statement

Classify the critical points of \(f(x,y) = 1-(x^2+y^2)^{1/3}\)

Solution

Thomas Wernau - 4365 video solution

video by Thomas Wernau

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Find the relative max and mins of \(f(x,y) = 2x^2+y^2+8x-6y+20\)

Problem Statement

Find the relative max and mins of \(f(x,y) = 2x^2+y^2+8x-6y+20\)

Solution

Thomas Wernau - 4366 video solution

video by Thomas Wernau

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Identify any extrema of the function \(f(x,y) = (x+3)^2 + (y-1)^2\) by recognizing its given form. Verify by using partial derivatives to locate any critical points and test for extrema.

Problem Statement

Identify any extrema of the function \(f(x,y) = (x+3)^2 + (y-1)^2\) by recognizing its given form. Verify by using partial derivatives to locate any critical points and test for extrema.

Hint

In the first few seconds of the video clip, he shows a plot of \(f(x,y)\).

Problem Statement

Identify any extrema of the function \(f(x,y) = (x+3)^2 + (y-1)^2\) by recognizing its given form. Verify by using partial derivatives to locate any critical points and test for extrema.

Hint

In the first few seconds of the video clip, he shows a plot of \(f(x,y)\).

Solution

Thomas Wernau - 4362 video solution

video by Thomas Wernau

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Find the absolute max and min values of the function on D where D is the enclosed triangular region with vertices \((0,0)\), \((0,2)\) and \((4,0)\) given \(f(x,y) = x+y-xy\)

Problem Statement

Find the absolute max and min values of the function on D where D is the enclosed triangular region with vertices \((0,0)\), \((0,2)\) and \((4,0)\) given \(f(x,y) = x+y-xy\)

Solution

Thomas Wernau - 4367 video solution

video by Thomas Wernau

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An electronics manufacturer determines that the profit (in dollars) by producing \(x\) units of Playstations and \(y\) units of X-Boxes is modeled by \(P(x,y) = 8x+10y-(0.001)(x^2+xy+y^2)-10000\). Find the production level that produces the maximum profit, making sure to show you find a maximum. What is the maximum profit?

Problem Statement

An electronics manufacturer determines that the profit (in dollars) by producing \(x\) units of Playstations and \(y\) units of X-Boxes is modeled by \(P(x,y) = 8x+10y-(0.001)(x^2+xy+y^2)-10000\). Find the production level that produces the maximum profit, making sure to show you find a maximum. What is the maximum profit?

Solution

Thomas Wernau - 4368 video solution

video by Thomas Wernau

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An open top box is to hold \(100 cm^3\) of a liquid. Find the equation of the surface area in terms of \(x\) and \(y\). Calculate the partial derivatives (gradient vector) and use this to find the values of \(x\) and \(y\) that would minimize the surface area. Make sure to show that you found a minimum.

Problem Statement

An open top box is to hold \(100 cm^3\) of a liquid. Find the equation of the surface area in terms of \(x\) and \(y\). Calculate the partial derivatives (gradient vector) and use this to find the values of \(x\) and \(y\) that would minimize the surface area. Make sure to show that you found a minimum.

Solution

Thomas Wernau - 4369 video solution

video by Thomas Wernau

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Really UNDERSTAND Calculus

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