Extrema In 3Space
For a full lecture on this topic, including optimization, here is a great video.
video by Prof Leonard 

Recommended Books on Amazon (affiliate links)  

In order to determine extrema of functions in 3space, 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.
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.
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.
video by Thomas Wernau 

Practice
Find the critical points of \(f(x,y)=xy\)
Problem Statement
Find the critical points of \(f(x,y)=xy\)
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
video by Thomas Wernau 

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Find the critical points of \(f(x,y) = 4+x^3+y^33xy\)
Problem Statement
Find the critical points of \(f(x,y) = 4+x^3+y^33xy\)
Solution
video by Thomas Wernau 

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Classify the critical points of \(f(x,y) = 4+x^3+y^33xy\)
Problem Statement
Classify the critical points of \(f(x,y) = 4+x^3+y^33xy\)
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
video by Thomas Wernau 

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Find the relative max and mins of \(f(x,y) = 2x^2+y^2+8x6y+20\)
Problem Statement
Find the relative max and mins of \(f(x,y) = 2x^2+y^2+8x6y+20\)
Solution
video by Thomas Wernau 

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Identify any extrema of the function \(f(x,y) = (x+3)^2 + (y1)^2\) 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 \(f(x,y) = (x+3)^2 + (y1)^2\) 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 \(f(x,y) = (x+3)^2 + (y1)^2\) 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
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 \(f(x,y) = x+yxy\)
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 \(f(x,y) = x+yxy\)
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 XBoxes 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 XBoxes 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
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
video by Thomas Wernau 

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Really UNDERSTAND Calculus
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