00:01
So in this question, it seems that you requested for us to do part a of this question, where we are determining the critical point of the function, 1 plus 2x plus 8y, minus x squared minus 2y squared, and we're going to determine whether that point is a max or a min.
00:21
So let's get our critical point.
00:23
Critical points are places where both partial derivatives are equal to 0.
00:28
So let's see, my partial of f with respect to x, that's going to be just 2 minus 2x.
00:40
While my partial of f with respect to y, that's going to be 8 minus 4 y.
00:48
And we need both of these partial derivatives to be equal to 0.
00:54
And so if we set both of these partial derivatives equal to 0, what are we getting? looks like we're getting an x coordinate of one, and it looks like we're getting a y coordinate of two.
01:14
And so those are the coordinates of my critical point.
01:21
And so we have a critical point this time at the point, 1 comma 2.
01:30
Now, if i want to know if this point is a maximum or a minimum, we're going to analyze the function d.
01:40
And you may recall that the function d is equal to the second partial of f with respect to x twice times the second partial of f with respect to y twice minus the mixed partial f sub xy quantity being squared.
02:04
And so let's see here.
02:06
What is my f subxx, xx.
02:09
I take the partial derivative with respect to x of 2 minus 2x, and that gives you negative 2.
02:19
Then i take the partial of f with respect to y twice...