6. Let f(x) be differentiable for a < x < b. Which of the following statements must be true?
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A. f is increasing on (a,b) |
B. f is continuous on (a,b) |
C. f is bounded on [a,b] |
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D. f is continuous on [a,b] |
E. f is decreasing on [a,b] |
Solution: The answer is B
Statements A and E are false, as f(x) = x2 on (-1, 1) shows. This function is decreasing on one part of its domain and increasing on another.
Statements C and D are false. As a counterexample consider f(x) = 1/x on the interval (0, 1).
(B) is a standard result in a first course in calculus.