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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.64 Optimization and Lagrange Multipliers

Keith Ball

1 Optimization

Soon after being introduced to calculus, most students learn of its application to optimization: that is, to the problem of finding the largest or smallest value of a given differentiable function, which is usually referred to as the objective function. A very helpful observation is that if f is an objective function that is maximized or minimized at x, then the tangent to the graph at the point (x, f(x)) will be horizontal, since otherwise we can find some value x′ close to x for which f(x′) is higher. This means that we can narrow down the search for the maximum and minimum values of f by looking just at the values of f(x) for which f′(x) = 0.

Now suppose that we have ...

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