Decide whether each conjecture is true or false. Provide an argument for those that are true and a counterexample for each one that is false. (a) If f' exists on an interval and is not constant, then...

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0, then there exists a d-neighborhood Vi(c) around c in which f'(x) > 0 for all r E Vs(c). (c) If f is differentiable on an interval containing zero and if lim, 0 f'(x) = L, then it must be that L = f'(0). "/>
Extracted text: Decide whether each conjecture is true or false. Provide an argument for those that are true and a counterexample for each one that is false. (a) If f' exists on an interval and is not constant, then f' must take on some irrational value. (b) If f' exists on an open interval and there is some point e where f"(c) > 0, then there exists a d-neighborhood Vi(c) around c in which f'(x) > 0 for all r E Vs(c). (c) If f is differentiable on an interval containing zero and if lim, 0 f'(x) = L, then it must be that L = f'(0).