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17.10 Determine whether the hypotheses of Rolle s theorem hold for each function f , and if they do, verify the conclusion of the theorem. (a) f (x) = x 2 2x 3 on [ 1, 3] (c) f (x) = 9x 3 4x on 2 , 2 3 3 (e) f (x) = x2 x 6 on [ 2, 3] x 1 (b) f (x) = x 3 x on [0, 1] (d) f (x) = x 3 3x 2 + x + 1 on [1, 1 + 2] x 3 2x 2 5x + 6 if x = 1 x 1 on [ 2, 3] (f ) f (x) = 6 if x = 1 x2 if 0 x 1 on [0, 2] (h) f (x) = 2 x if 1 < x 2
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(g) f (x) = x 2/3 2x 1/3 on [0, 8]
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17.11 Verify that the hypotheses of the mean-value theorem hold for each function f on the given interval, and nd a value c satisfying the conclusion of the theorem. (a) f (x) = 2x + 3 on [1, 4] (c) f (x) = x 3/4 on [0, 16] (e) f (x) = 25 x 2 on [ 3, 4] (b) f (x) = 3x 2 5x + 1 on [2, 5] x+3 on [1, 3] (d) f (x) = x 4 1 on [0, 2] (f ) f (x) = x 4
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THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE
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17.12 Determine where the function f is increasing and where it is decreasing. Then sketch the graph of f . (a) f (x) = 3x + 1 (d) f (x) = 1 4x x 2 (g) f (x) = x 3 9x 2 + 15x 3 (b) f (x) = 2x + 2 (e) f (x) = 1 x2 1 (h) f (x) = x + x (c) f (x) = x 2 4x + 7 1 (f ) f (x) = 9 x2 3 (i) f (x) = x 3 12x + 20
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17.13 Let f be a differentiable function such that f (x) = 0 for all x in the open interval (a, b). Prove that there is at most one zero of f (x) in (a, b). [Hint: Assume, for the sake of contradiction, that c and d are two zeros of f , with a < c < d < b, and apply Rolle s theorem on the interval [c, d].] 17.14 Consider the polynomial f (x) = 5x 3 2x 2 + 3x 4. (a) Show that f has a zero between 0 and 1. (b) Show that f has only one real zero. [Hint: Use Problem 17.13.] 17.15 Assume f continuous over [0, 1] and assume that f (0) = f (1). Which one(s) of the following assertions must be true (a) If f has an absolute maximum at c in (0, 1), then f (c) = 0. (b) f exists on (0, 1). (c) f (c) = 0 for some c in (0, 1). (d) lim f (x) = f (c) for all c in (0, 1).
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(e) f has an absolute maximum at some point c in (0, 1). 17.16 Let f and g be differentiable functions. (a) If f (a) = g(a) and f (b) = g(b), where a < b, show that f (c) = g (c) for some c in (a, b). (b) If f (a) g(a) and f (x) > g (x) for all x, show that f (x) > g(x) for all x > a. (c) If f (x) > g (x) for all x, show that the graphs of f and g intersect at most once. [Hint: In each part, apply the appropriate theorem to the function h(x) = f (x) g(x).] 17.17 Let f be a differentiable function on an open interval (a, b). (a) If f is increasing on (a, b), prove that f (x) 0 for every x in (a, b). f (x + h) f (x) and Problem 9.10(a) applies. Hint: f (x) = lim + h h 0 (b) If f is decreasing on (a, b), prove that f (x) 0 for every x in (a, b). 17.18 The mean-value theorem predicts the existence of what point on the graph of y = 3 x between (27, 3) and (125, 5)
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17.19 (Generalized Rolle s Theorem) Assume f is continuous on [a, b] and differentiable on (a, b). If f (a) = f (b), prove that there is a point c in (a, b) such that f (c) = 0. [Hint: Apply Rolle s theorem to g(x) = f (x) f (a).] 17.20 Let f (x) = x 3 4x 2 + 4x and g(x) = 1 for all x. (a) Find the intersection of the graphs of f and g. (b) Find the zeros of f . (c) If the domain of f is restricted to the closed interval [0, 3], what would be the range of f 17.21 Prove that 8x 3 6x 2 2x + 1 has a zero between 0 and 1. [Hint: Apply Rolle s theorem to the function 2x 4 2x 3 x 2 + x.] 17.22 Show that x 3 + 2x 5 = 0 has exactly one real root.
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