# Swim, Swim and Walk, or Walk?

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A lifeguard at the edge of a circular pool sees a baby fall into the water at another point on the edge of the pool. To save the baby, should he swim directly to her, swim to some point and then walk, or walk all the way around to her? (He has to walk, and not run, because the edge of the pool is slippery.)

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Contributed by: Roger B. Kirchner (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

Former Harvard Professor J. L. Walsh emphasized rigor in solving max-min problems. In this case, the principles are:

A function continuous on a closed bounded interval has a (global) minimum there.

A function differentiable at cannot have a minimum there unless .

Consequently, a function continuous on has a miminum value, and the minimum can occur only at an endpoint or where or where does not exist.

J. L. Walsh, *A Rigorous Treatment of Maximum-Minimum Problems in the Calculus*, Boston: Heath, 1962.

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