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MindTouch
http://mindtouch.com

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# Lagrange Multipliers

Many applied max/min problems take the form of the last two examples: we want to find an extreme value of a function, like $$V=xyz$$, subject to a constraint, like $$1=\sqrt{x^2+y^2+z^2}$$. Often this can be done, as we have, by explicitly combining the equations and then finding critical points. There is another approach that is often convenient, the method of Lagrange multipliers.

It is somewhat easier to understand two variable problems, so we begin with one as an example. Suppose the perimeter of a rectangle is to be 100 units. Find the rectangle with largest area. This is a fairly straightforward problem from single variable calculus. We write down the two equations: $$A=xy$$, $$P=100=2x+2y$$, solve the second of these for $$y$$ (or $$x$$), substitute into the first, and end up with a one-variable maximization problem.

Let's now think of it differently: the equation $$A=xy$$ defines a surface, and the equation $$100=2x+2y$$ defines a curve (a line, in this case) in the $$x$$-\)y\) plane. If we graph both of these in the three-dimensional coordinate system, we can phrase the problem like this: what is the highest point on the surface above the line? The solution we already understand effectively produces the equation of the cross-section of the surface above the line and then treats it as a single variable problem. Instead, imagine that we draw the level curves (the contour lines) for the surface in the $$x$$-$$y$$ plane, along with the line.