We begin with a definition which is illustrated in

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We begin with a definition which is illustrated in Figure 4. DEFINITION Let / be a function, and A a set of numbers contained in the domain of /. A point x in A is a local maximum [minimum] point for / on A if there is some <5 > such that x is a maximum [minimum] point for / on A nix -8,x + 8). THEOREM 2 If x is a local maximum or minimum for / on (a, b) and / is differentiable at x, then fix) = 0. proof You should see why this is an easy application of Theorem 1 . |
1 90 Derivatives and Integrals The converse of Theorem 2 is definitely not true it is possible for f'(x) to be even if x is not a local maximum or minimum point for /. The simplest example is provided by the function f(x) x 3 ; in this case f'(0) = 0, but / has no local maximum or minimum anywhere. Probably the most widespread misconceptions about calculus are concerned with the behavior of a function / near x when f'(x) = 0. The point made in the previous paragraph is so quickly forgotten by those who want the world to be simpler than it is, that we will repeat it: the converse of Theorem 2 is not true the condition f'(x) does not imply that x is a local maximum or minimum point of /. Precisely for this reason, special terminology has been adopted to describe numbers x which satisfy the condition fix) 0. DEFINITION A critical point of a function / is a number x such that f'(x) = 0. The number f(x) itself is called a critical value of /. x\ a local minimum point FIG 1 RE 4 X2 a local maximum point The critical values of /, together with a few other numbers, turn out to be the ones which must be considered in order to find the maximum and minimum of a given function /. To the uninitiated, finding the maximum and minimum value of a function represents one of the most intriguing aspects of calculus, and there is no denying that problems of this sort are fun (until you have done your first hundred or so). Let us consider first the problem of finding the maximum or minimum of / on a closed interval [a,/?]. (Then, if / is continuous, we can at least be sure that a maximum and minimum value exist.) In order to locate the maximum and minimum of / three kinds of points must be considered: (1) The critical points of / in [a, b]. (2) The end points a and b. (3) Points x in [a, b] such that / is not differentiable at x. If x is a maximum point or a minimum point for / on [a, £>] , then x must be in one of the three classes listed above: for if x is not in the second or third group, then x is in (a, b) and / is differentiable at x; consequentiy f'(x) = 0, by Theorem 1, and this means that .v is in the first group. If there are many points in these three categories, finding the maximum and minimum of / may still be a hopeless proposition, but when there are only a few critical points, and only a few points where / is not differentiable, the procedure is fairly straightforward: one simply finds f(x) for each x satisfying f'(x) = 0, and f(x) for each x such that / is not differentiable at x and, finally, f{a) and fib).

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