Schaum's Outline of Theory and Problems of Beginning Calculus

h,CHAP. 171 THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE 135(c) f’(c) = 0 for some c in (0, 1).(d) lim f(x) =f(c) for all c in (0, 1).x-c(e) fhas an absolute maximum at some point c in (0, 1).17.1617.1717.1817.1917.20Letfand g be differentiable functions.(a) Iff(a) = g(a) andf(b) = g(b), where a < b, show thatf’(c) = g’(c) for some c in (a, b).(b) Iff(a) 2 g(a) andf’(x) > g‘(x) for all x, show thatf(x) > g(x) for all x > a.(c) If f’(x) > g’(x) for all x, show that the graphs off and g intersect at most once. [Hint: In each part,apply the appropriate theorem to the function h(x) =f(x) - g(x).]Let f be a differentiable function on an open interval (a, b).(a) Iffis increasing on (a, b), prove that f’(x) 2 0 for every x in (a, b).[Hint:+ f’(x) = lim f(x -f(x) and Problem 9.1qa) applies.h-O+ h1(b) Iffis decreasing on (a, b), prove thatf’(x) 5 0 for every x in (a, b).The mean-value theorem predicts the existence of what point on the graph of y = fi between (27, 3) and(125, 5)?(Generalized Rolle’s Theorem) Assumefis continuous on [a, b) and differentiable on (a, b). Iff(a) =f(b),prove that there is a point c in (a, b) such thatf’(c) = 0. [Hint: Apply Rolle’s theorem to g(x) =f(x) -f(a).]Letf(x) = x3 - 4x2 + 4x and g(x) = 1 for all x.(a) Find the intersection of the graphs offand g.(b) Find the zeros oft(c) If the domain offis restricted to the closed interval [0, 31, what would be the range off?17.2117.2217.2317.2417.2517.26Prove that 8x3 - 6x2 - 2x + 1 has a zero between 0 and 1. [Hint: Apply Rolle’s theorem to the function2x4 - 2x3 - x2 + x .~Show that x3 + 2x - 5 = 0 has exactly one real root.Prove that the equation x4 + x = 1 has at least one solution in the interval CO, 13.Find a point on the graph of y = x2 + x + 3, between x = 1 and x = 2, where the tangent line is parallel tothe line connecting (1, 5) and (2,9).(a) Show thatf(x) = x5 + x - 1 has exactly one real zero.(b) Locate the real zero of x5 + x - 1 correct to the first decimal place.(a) Use a graphing calculator to estimate the intervals in which the functionf(x) = x4 - 3x2 + x - 4is increasing and the intervals in which it is decreasing.(b) As in part (a), but for the functionf(x) = x3 - 2x2 + x - 2.