Larange Multiplier Problem!?

Question

So I was going through the old exam papers and found this question. I tried to look for a something a bit similar to this but found nothing. It was hinted that a similar question might appear in this year's exam so if anybody can help me then it'd be greatly appreciated. The question is as follows:
Use the method of Lagrange multipliers to find a relative maximum of the function f(x,y,z)=xyz, subjects to the constraints: x+y+z=4 and x-y-z=3.

Answer 1

OK, I'm going to leave out some details, but here's the way it goes. When you want to maximize something, but you have conditions on the variables, then for each condition (that is to say, for each equation dictating a constraint) introduce a new variable. We're going to call them p and q in our example, but typically they are denoted by lambda sub 1, lambda sub 2, etc. Then the new function we're going to maximize/minimize (find extrema points for) will be given by:

F = xyz + p(x+y+z-4) + q(x-y-z-3)

Clearly, when you take a partial with respect to p (and set it equal to 0) you get your original condition 1 back and when you take a partial with respect to 2 (and set it equal to 0) you get your original condition 2 back. So the additional parts don't contribute any extra value to the function, but the variables are now tied together. If you are lucky, you'll be able to solve the system of equations.

So, let's take partials with respect to x, y, and z and set them equal to 0:
yz + p + q = 0
yx + p - q = 0
xz + p - q = 0

Subtracting the last two equations, we arrive at: x(y-z) = 0. In other words, x = 0 or y-z = 0. But if you stuff x=0 into the original constraints, you see that you get a contradiction. Therefore, y=z

Furthermore, if you subtract the two original conditions from each other, you see that x = 7/2.

Plugging this into the first condition, we get: y+z = 1/2
And from y=z we have
y = z = 1/4

At this point, you should check boundary points (when applicable) and whether your solution makes sense.

In any case, this function takes a relative maximum at:
(7/2, 1/4, 1/4)