00:01
This question gives us a function of two variables and then has us used theorem 7 .2 to find and classify all of the function's critical points.
00:10
The function we have is f of xy is equal to 2x squared plus y cubed.
00:18
I'm not sure why i put it cubed there, minus x squared y minus 3y.
00:24
We want to look for the critical points, which are points where both partial derivatives are equal to zero.
00:31
And so we have to calculate the partial derivatives first, fx.
00:34
We have 2x squared, which derivative would just be 4x.
00:38
This y cubed and this negative 3y will both be treated as constant, and so they will have a derivative of 0.
00:44
The only last term to focus on is the negative x squared y, which will have derivative negative 2xy, treating y as a constant.
00:53
Now we look at the partial derivative of f with respect to y.
00:56
So all of the only x terms, like 2x squared, are going to have a partial derivative of 0.
01:05
And so we have y cubed, which has derivative 3y squared, and then negative x squared y, which would just be negative x squared, then minus 3.
01:16
So we want to set these both equal to zero and each other.
01:22
And we want to try to figure out which points will work.
01:29
And so we'll go ahead and just try to solve one of these equations.
01:33
One thing to note is in this equation here, if x is equal to zero, it will always be zero...