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Lagrange multipliers are simply means of introducing constraints into an optimization problem. Perhaps the easiest way to think of this is to imagine two surfaces intersecting, and to ask what the largest value of the first function is upon the curve of intersection.
Let's examine Figure 1 on page 981, to get the idea.
The cool thing is that we can introduce a new variable (the Lagrange multiplier), increasing the dimension of our problem; and then solve that higher-dimensional problem using unconstrained optimization (which we just did -- finding maxes and mins).
So we cast the problem into a higher dimension to solve it.
Problems:
I think that you'll enjoy this approach more than the previous approach from Tuesday!
But I could have made the solution come a little quicker: after using symmetry and Nathan's good suggestion, we arrived at \[ (l-w)(32-lw)=0 \] from which we can conclude that $l=w$ (since $lw=32$ implies that $h=0$). Then we plug that back into one of the two equations to get $32-3w^2=0$, so, in the end (consistent with the symmetry), \[ l=w=h=\sqrt{\frac{32}{3}} \]
