Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 17) THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE 131‘P1 IA Y*a b XFig. 17-4Corollary 17.5:Iffis a continuous function with domain [a, b], then the range offis either a closedinterval or a point.Solved Problems17.1 Verify Rolle’s theorem forf(x) = x3 - 3x2 - x + 3 on the interval [l, 31.fis differentiable everywhere and, therefore, also continuous. Furthermore,f(i) = ( 93 - 3(1)2 - 1 + 3 = 1 - 3 - 1 + 3 = of(3) = (3)3 - 3(3)2 - 3 + 3 = 27 - 27 - 3 + 3 = 0so that all the hypotheses of Rolle’s theorem are valid. There must then be some c in (1, 3) for whichf ‘(c) = 0.Now, by the quadratic formula, the roots off’(x) = 3x2 - 6x - 1 = 0 areConsider the root c = 1 + 4fi. Since fi < 3,1