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More Maximum and Minimum Problems
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Until now we have been able to nd the absolute maxima and minima of differentiable functions only on closed intervals (see Section 14.2). The following result often enables us to handle cases where the function is de ned on a half-open interval, open interval, in nite interval, or the set of all real numbers. Remember that, in general, there is no guarantee that a function has an absolute maximum or an absolute minimum on such domains. Theorem 24.1: Let f be a continuous function on an interval I , with a single relative extremum within I . Then this relative extremum is also an absolute extremum on I . Intuitive Argument: Refer to Fig. 24-1. Suppose that f has a relative maximum at c and no other relative extremum inside I . Take any other number d in I . The curve moves downward on both sides of c. Hence, if the value f (d) were greater than f (c), then, at some point u between c and d, the curve would have to change direction and start moving upward. But then f would have a relative minimum at u, contradicting our assumption. The result for a relative minimum follows by applying to f the result just obtained for a relative maximum.
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For a rigorous proof, see Problem 24.20.
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Fig. 24-1
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MORE MAXIMUM AND MINIMUM PROBLEMS
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EXAMPLES
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(a) Find the shortest distance from the point P(1, 0) to the parabola x = y2 [see Fig. 24-2(a)].
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Fig. 24-2
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