00:01
We were asked to minimize the function, x times y times z, subject to two constraints.
00:11
Both of these are hyperplanes in this x, y, z space.
00:17
So we have x plus 3y minus 6 equals 0 and x minus 2 z equals 0.
00:23
Those are two constraints.
00:24
So we need to form our function here, our supplemental function here, is f plus lambda 1, our first lagrange multiplier times our first constraint plus lambda 2, our second lgrange multiplier times our second constraint.
00:42
And now this thing is now a function of x, y, z, lambda 1, and lambda 2.
00:47
And we need to take the partials with respect to all of those and set them all equal to 0.
00:53
So partial with respect to x is yz plus lambda 1 plus lambda 2.
00:57
Respect to y is xz plus 3 lambda 1.
01:01
Respect to z is xy minus 2 lambda 2.
01:05
And then the partials with respect to the constraints just give us our constraint equations back.
01:12
So now we just need to, you know, we've got five equations and five unknowns.
01:17
They're not linear, but you can basically eliminate the lambdas and, you know, go through and figure out what, you know, just keep back substituting.
01:31
And in the end, what we get, is we get two solutions.
01:36
One is 0 to 0...