Schaum's Outline of Theory and Problems of Beginning Calculus

106 MAXIMUM AND MINIMUM PROBLEMS [CHAP. 14(6)Fig. 14-4Definition: A critical number of a functionfis a number c in the domain off for which eitherf’(c) = 0orf’(c) is not defined.EXAMPLES(a)Let f(x) = 3x2 - 2x + 4. Then f‘(x) = 6x - 2. Since 6x - 2 is defined for all x, the only critical numbers aregiven byThus, the only critical number is 3.6~-2=06x = 2X=$=i(b) Letf(x) = x3 - x2 - 5x + 3. Thenf’(x) = 3x2 - 2x - 5, and since 3x2 - 2x - 5 is defined for all x, the onlycritical numbers are the solutions of3x2 - 2x - 5 = 0(3x - 5)(x + 1) = 03x-5=0or x+1=03x = 5 or x= -1x=3 or x- -1Hence, there are two critical numbers, - 1 and 3.We already know from the example in Section 13,l that f’(0) is not defined. Hence, 0 is a critical number.Since D,(x) = 1 and DA -x) = - 1, there are no other critical numbers.Method for Finding Absolute ExtremaLet f be a continuous function on a closed interval [a, b]. Assume that there are only a finitenumber of critical numbers c1, c2, . . . , ck off inside [a, b]; that is, in (a, b). (This assumption holds formost functions encountered in calculus.) Tabulate the values off at these critical numbers and at theendpoints U and b, as in Table 14-1. Then the largest tabulated value is the absolute maximum off on[a, b], and the smallest tabulated value is the absolute minimum off on [a, b]. (This result is proved inProblem 14.1.)