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In Problems 116, use the Intermediate Value Theorem to show that the function has at least one zero in the given interval. In Problems 1730, determine whether or not f' has a zero in the interval (a, b). Warning: Rolle's Theorem may give a wrong answer unless all the hypotheses are met. In Problems 3542, find a point c in (a, b) such that f(b)  f(a) = f'(c)(b  a). 43 Use Rolle's Theorem to show that the function f(x) = x^{3}  3x + b cannot have more than one zero in the interval [1, 1], regardless of the value of the constant b. 44 Suppose f, f', and f" are all continuous on the interval [a, b], and suppose f has at least three distinct zeros in [a, b] Use Rolle's Theorem to show that f" has at least one zero in [a, b] 45 Suppose that f"(x) > 0 for all real numbers x, so that the curve y = f(x) is concave upward on the whole real line as illustrated in the figure. Let L be the tangent line to the curve at x = c. Prove that the line L lies below the curve at every point x ≠ c.


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