00:01
In order to optimize the objective function f here, given the constraint function, using lagrange multipliers, we'll go ahead and first take the partial with respect to x here of the f function, and that would simply give us one because the derivative of x would just be one and y would be treated as a constant.
00:22
Similarly, the partial with respect to y would just be one, that derivative.
00:27
Switching over to the objective function, the partial with respect to x would be 2x, and the partial with respect to y would be 2y.
00:37
And then we're going to use the three steps that the authors give us here to set up three functions or three systems of equations that will set up.
00:48
So the first one is the partial.
00:52
We're going to set this function equal to this one with a lagrange multiplier.
00:58
In front of it.
01:00
So we're going to have one equals the lagrange multiplied by 2x.
01:13
Similarly, we'll do this partial with respect to y is going to equal lagrange multiplied by 2y.
01:24
1 equals lagrange multiplied by 2y.
01:32
And then lastly, our third constraint is that we're going to be.
01:38
Going to use the g function that we're given.
01:43
So we'll go ahead and write that in blue here.
01:46
And then at this point, we want to find a way to solve.
01:51
We could do this a lot of different ways.
01:53
We could solve for the lagrange multiplier.
01:56
We could solve for x or for y.
01:58
So i've chosen to use equations one and two and basically just set them equal to each other, right? and so in other words, if we moved the 2x over to the other side and then the 2y over to the other side, we would have 1 over 2x is equal to the lagrange multiplier, which if we then substitute that for the second equation, that would just be equal to 2y...