Schaum's Outline of Theory and Problems of Beginning Calculus

Chapter 24More Maximum and Minimum ProblemsUntil now we have been able to find the absolute maxima and minima of differentiable functionsonly on closed intervals (see Section 14.2). The following result often enables us to handle cases wherethe function is defined on a half-open interval, open interval, infinite interval, or the set of all realnumbers. Remember that, in general, there is no guarantee that a function has an absolute maximum oran absolute minimum on such domains.Theorem 24.1: Let f be a continuous function on an interval 9, with a single relative extremum within9. Then this relative extremum is also an absolute extremum on 9.Intuitive Argument: Refer to Fig. 24-1. Suppose that f has a relative maximum at c and no otherrelative extremum inside 3. Take any other number d in 9. The curve moves downwardon both sides of c. Hence, if the valuef(d) were greater thanf(c), then, at somepoint U between c and d, the curve would have to change direction and start movingupward. 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 obtainedfor a relative maximum.For a rigorous proof, see Problem 24.20.EXAMPLES1 1 1*C U dFig. 24-1(a) Find the salortest 4 istance from the point P(1,O) to the parabola x = y2 [see Fig. 24-2(a)].The distance from an arbitrary point Q(x, y) on the parabola to the point P(1,O) is, by (24,U=J-= d- [y2 = x at Q]=Jx2-2X+l+x=J-But minimizing U is equivalent to minimizing u2 = F(x) = x2 - x + 1 on the interval CO, + co) (the value of xis restricted by the fact that x = y2 2 0).The only critical number is the solution ofF(x) = 2x - 1 F”(x) = 2F(x) = 2x - 1 = 0or x = 3Now F”(3) = 2 > 0. So by the second-derivative test, the function F has a relative minimum at x = 3. Theorem24.1 implies that this is an absolute minimum. When x = 3,y2=x=- 1 and y=+-= 1 .4 f- 42-$ +JzJz= 2Thus, the points on the parabola closest to (1,O) are (4, &2) and (4, -4/2).179