Tham khảo tài liệu 'stewart - calculus - early transcendentals 6e hq (thomson, 2008) episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 274 Illi CHAPTER 4 APPLICATIONsOF DIFFERENTIATION We have shown that f8fC 0 and also that f8fC 0. Since both of these inequalities must be true the only possibility is that Ặc 0. We have proved Fermat s Theorem for the case of a local maximum. The case of a local minimum can be proved in a similar manner or we could use Exercise 76 to deduce it from the case we have just proved see Exercise 77 . The following examples caution US against reading too much into Fermat s Theorem. We can t expect to locate extreme values simply by setting f8ịx 0 and solving for X. FIGURE 9 If f _r A 3 then f 0 0 but f has no maximum or minimum. EXAMPLE 5 If x X3 then f8fX 3x2 so 0 0. But f has no maximum or minimum at 0 as you can see from its graph in Figure 9. Or observe that X3 0 for X 0 but X31 0 for X 1 0. The fact that 0 0 simply means that the curve y X3 has a horizontal tangent at 0 0 . Instead of having a maximum or minimum at 0 0 the curve crosses its horizontal tangent there. EXAMPLE 6 The function x IXI has its local and absolute minimum value at 0 but that value can t be found by setting Ặx 0 because as was shown in Example 5 in Section Ặ0 does not exist. See Figure 10. FIGURE 10 If f x IA . then f 0 0 is a minimum value but f 0 does not exist. WARNING Examples 5 and 6 show that we must be careful when using Fermat s Theorem. Example 5 demonstrates that even when Ặc 0 there need not be a maximum or minimum at c. In other words the converse of Fermat s Theorem is false in general. Furthermore there may be an extreme value even when Ặc does not exist as in Example 6 . Fermat s Theorem does suggest that we should at least start looking for extreme values of at the numbers c where Ặc 0 or where Ặc does not exist. Such numbers are given a special name. 6 DEFINITION A critical number of a function is a number c in the domain of such that either Ặc 0 or Ặc does not exist. Figurd IshowsagraphDfthefunctioi in Example. It supporteuranswebecause therdsahorizontdfengentvhenv .
