Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Find the extrema of the function $f(x, y)=x^{2}+y^{2}-4 x-4 y+28$ defined over the region bounded by $y=0, x=0$ and $x+y=8$

$$\operatorname{Max}=60, \min =20$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

Johns Hopkins University

Campbell University

Oregon State University

Baylor University

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

06:45

Find the absolute extrema …

Hi there. In this video, we're going to find the absolute maximum on the absolute minimum values of the function. F of X Y is equal to X squared plus y squared minus two y plus one subject to the circular domain x y Such the X squared plus y squared is less than or equal to four. I've drawn this domain here, Onda and the absolute maximum An absolute minimum values off the function will either be a critical point off the ah function within the domain or on the boundary of the domain. So first, we're going to calculate the critical points, calculate the value of F at these critical points and then again to calculate the value of F on the boundary. Um on de fined the maximum minimum. Put this by poetic ordinate characterization. So first we look for the critical points of F. So the partial derivative of F with respect to X is just equal to two X setting. This equal to zero tells us that X is equal to zero. The partial derivative with respect why is equal to two y minus two secular equal to zero tells us that why is equal to one Serve the 0.1 as a critical point of this function. If we can't let f of 01 this is zero squared plus one squared minus to plus one. Andi, this is equal Teoh zero. So we've exhausted the critical points. It's now we have to look at the values on the boundary. So we look at the boundary. We noticed that this is a circle of radius to since a circle is X squared, plus y squared is equal to R squared on the entire M in a domain is X squared plus y squared is, um, less unequal to ask red, which is the case we have have the boundary is when it's equal to R squared. In our case, this is four, which is two squared. We can Therefore, Paramount tries the boundary by saying X is equal to to co sign a t uh, where t um goes from nought to a cheap I on Why is equal to to sign t. So we put this in, um, f of tea now. Um, So instead of calling this f, let's call this G of T is equal f of X of T why of tea, which is equal Teoh to cause t squared plus two scientific or squared minus two times to sign T plus one This we have four times coastline square, T plus sign it squared T and minus four sine of t plus one. This gives us one. So we have five minus four. Sign of T. Now, this is a sign of tea. How that this is minus sine of t which we're dealing with. So in fact, we just need to deal with a sign of t. So we're looking for the maximum and minimum off this the, uh, maximum. Um, off. This is achieved when, uh ah, it's all dependent on science. T so sign T is bounded between one on minus one. So when t is equal to, uh, Piper to we get our maximum when t is equal to minus five a two. We rich Eat it, Eva. Minimum G off pi over two is ah, five minus four is equal to one G of minus pi over two is equal to five plus four is equal to nine. So these are the maximum minimum points off our boundary. Notice that one is still larger than the value zero. We got a critical point. So you find that our minimum value a 1,000,000 value is zero on our on. This is at the 0.1 and ah, maximum value, which is on the boundary, is nine. And this is when t is equal. Teoh minus pi over two. And in fact, his outside of our choice. So instead of using my ass paper to we use three pilot too. And we achieved the same result colleges when t is equal to three pi over two. So this is when X is equal to two. Co sign of three pi over two. And why is equal to to sign three pilot, too? Uh, co sign of three part 20 sign of three pi over two to minus one. So this is at, um what about by two. So at the 20.0 minus two. So you've found maximum and minimum values off f restricted to this circular domain. Thank you.

View More Answers From This Book

Find Another Textbook

02:00

(a) determine the average value of the given function over the given interva…

02:54

Use the properties about odd and even functions to evaluate the given integr…

03:20

Given $\int(2 x+3)^{2} d x,$ (a) evaluate this integral by first multiplying…

01:50

A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)…

01:44

Evaluate the given integral and check your answer.$$\int\left(t^{2}+1\ri…

01:17

Evaluate the given integral and check your answer.$$\int 2 e^{t} d t$$

03:40

For $f(x, y, z)=2 x^{2}-16 x+3 y^{2}-24 y+4 z^{2}-40 z+25,$ find the point(s…

01:30

Sketch the area represented by the given definite integral.$$\int_{-1}^{…

02:21

Evaluate the given integral.$$\int_{2}^{3} \int_{2 y}^{y^{2}}\left(x^{2}…

01:57

Determine the area of the indicated region.Region bounded by $f(x)=\sqrt…