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Select the statement(s) that are true:
(a) If the derivative f'(a) exists and is finite at a point, then f must be continuous at x = a.
(b) The function f(x) = 3tan(5x) - 7 is differentiable at every point where it is defined.
- (a) only
- (b) only
- Both (a) and (b)
- Neither (a) nor (b)
Correct answer: Both (a) and (b)
Solution
(a) If f'(a) exists and is finite, differentiability implies continuity, so f is continuous at a. (b) f(x)=3tan(5x)-7 is built from tan, which is differentiable everywhere it is defined; so it is differentiable on its whole domain. Both true -> 'Both (a) and (b)'.
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