00:01
In this problem, we want to use the lagrange multiplier method to find the extreme points of the given function f subjected to a certain condition.
00:11
So consider the two -dimensional function f of xy equal to x squared plus y squared, and the constraint g of xy equal to x minus y plus 6.
00:30
According to the lagrange multiplier method, we want to solve for when the gradient of f is equal to lambda times the gradient of g.
00:55
Where the gradient of f is equal to 2xi plus 2yj, and the gradient of g is equal to i minus j.
01:10
J.
01:13
And when we're equating vector equations, we need to be careful to equate the proper components.
01:22
So let's equate the x component and the y component.
01:24
So this is going to give us a set of two equations.
01:27
We have 2x is equal to lambda, and 2y is equal to minus lambda.
01:35
And of course, we also have the constraint that x minus y plus 6 is equal to 0.
01:46
So now we have a system of three equations and three unknown variables, x, y, and lambda.
01:53
Well, lambda is called the lagrange multiplier, so we don't need to explicitly solve for lambda, but it can certainly help to solve for x and y.
02:06
So if we add up our first two equations, we have that x plus y must be equal to 0...