TITLE: How to explain lagrange multipliers to a lay audience?
QUESTION [8 upvotes]: So I will be giving a seminar to a scientifically mature lay audience (think bio/social science undergrad level). I have been told that I should count on less than half the audience to have experience with calculus. I think I can explain the basic optimization of $f(x)$ pictorially and introducing (and for some re-introducing) the concept of a derivative. Then I plan to go on to Lagrange multipliers and discrete time optimal control. I don't want to lose them though, so I want to know if anyone has any good explanations, analogies, videos, graphics (or anything else) that will keep a lay audience engaged. Note I will use Lagrange multipliers in the talk to explain a particular application, but I want to convey the message that the math is interesting in and of itself and that it can be used for a variety of things. I'm less concerned about the details being correct than getting the main message across. This portion of the seminar should be less than 15 minutes so I don't have that much time.

REPLY [14 votes]: If you just need to convey the basic intuition about the use of Lagrange multipliers in optimization and you are really worried about keeping your audience engaged, you might want to rely on their every-day intuition about level curves using this kind of drawing 

(here is the .svg of the picture if you want to generate variants of it. You should probably edit it with Latexdraw 2.0 if you want to be able to alter the path, the gradient, or anything which is not in the background)

You start by giving them the intuition that "you are at a critical point only if the path you are walking on does not cross a level curve, but rather is tangent to one". 
Then you explain how this can be checked using gradients.
Finally, you try to make them understand that this is equivalent to the existence of a Lagrange multiplier. 

People might be a little puzzled by the last step. But you can hope that most of them will catch the two first steps.  Even if they don't get the last one, they may remember that the existence of the multiplier is a convenient way to check that we are at a critical point based on the algebraic formulation of the function.