8. Let f be a real-valued function defined on the closed interval [a,b]. Which of the following conditions guarantees the existence of a number c such that a < c < b and f(c) = 0 ?
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A. f is continuous on [a,b], and f(a) = f(b). |
B. f is differentiable on [a,b], and f’(a) = f’(b). |
C. f is continuous on [a,b], and f(a) and f(b) have opposite signs. |
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D. f is differentiable on [a,b], and f’(a) and f’(b) have opposite signs. |
E. f(a) = f(b), and f’(a) = f’(b). |
Solution: The answer is C
By the Intermediate Value Theorem, a continuous function on a closed interval [a, b] takes on all values between is minimum and maximum on that interval. Thus if f(a) and f(b) have opposite signs then 0 is an intermediate value; there is a c in (a, b) such that f(c) = 0.