## 17Calculus - 3-Space Extrema

##### 17Calculus

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

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.