00:01
In this problem, we want to find the absolute minimum and maximum of the following function on the given region.
00:09
So we have f of x, y equal to 2x cubed plus y to the power of 4, for x squared plus y squared less or equal than 64.
00:23
So let's start by trying to find critical points within the boundary.
00:36
I'm not going to put a question mark because i'm just going to look for critical points as we typically do without a constraint, and we're just going to verify if they're within the boundary.
00:50
So to find a critical point, we want to find locations where the first and second derivative of our function is equal to zero.
00:59
So let's calculate fx, our partial derivative of f with respect to x, this will give us 6x squared.
01:11
Next, fy, this will give us 4y cubed.
01:16
So our critical point will correspond to location where both partial derivatives are simultaneously equal to zero, and we see that our critical point corresponds to the point x equal to zero and y equal to zero, which is well within our boundary here.
01:45
So this satisfies x squared plus y squared less or equal than 64.
01:55
So now we found a critical point, and let's see if we have other critical points or other extreme points on our boundary.
02:14
So to solve for critical points on our boundary, we're going to maximize the function f given the constraint that x squared plus y squared is equal to 64.
02:34
So now we have, let's define our condition g of xy as x squared plus y squared minus 64, that is equal to zero.
02:54
So like before, let's calculate our first and second derivative of our boundary equation with respect to x.
03:01
So we're going to try to maximize f given this constraint using lagrange multipliers, and we're going to need the two quantities.
03:08
So gx will give us 2x and gy will give us 2y.
03:14
So now we want to find critical points when the gradient of f is equal to lambda times the gradient of g.
03:37
So we will have three equations...