00:01
So we're going to focus on the first problem here, which is g of x, y, equals x to the fourth, plus y to the fourth, plus 2x squared, y squared, minus 2x squared plus y cubed.
00:15
So the first thing we want to do is find our partial derivatives, so g sub x is going to be 4x cubed plus 4x squared minus 4x.
00:26
We're going to set that equal to zero and we get x equals zero and x squared plus y squared equals one then we take partial derivative with respect to y we get four y cubed plus four x squared y plus three y squared equal to zero we get y equal zero and four uh y squared plus four x squared equals three y so then based on this if we plug in x equals 0 then we're going to get y equals 0 and y equals 3 4th if we plug in y equals 0 to the other equation the x squared plus y squared equals 1 we'll get x equals plus or minus 1 so this ends up giving us the solutions 0 0 0 0 3 4ths uh negative 1 0 and 1 0 then considering our other equation x squared plus y squared equals one.
01:25
Substituting that into this one here, we end up with x squared equals 1 minus four thirds cubed, which is less than zero, so that's not possible.
01:38
So these right here are going to be our critical points.
01:41
Then we just want to test them all.
01:43
We're going to take g sub xx, which gives us 12x squared plus 4 y squared minus 4.
01:50
Gyy gives us 12 y squared plus 4x squared plus 4x squared plus 4x squared plus 6y, and then gxy gives us 8xy...