TRUE OR FALSE: The Mean Value Theorem (MVT) implies Rolle’s Theorem.

Question

TRUE OR FALSE: The Mean Value Theorem (MVT) implies Rolle’s Theorem.

Answer 1

Normally one proves MVT using rolles theorem. That said, if you assume MVT, then rolles theorem is also true.

Pick a function f continuous on an interval [a,b]. Also assume f is differentiable in (a,b). The mean value theorem says that there exists some c in (a, b) such that f ' (c) = [f(b) - f(a)]/(b-a).

Suppose also that f(a) = f(b), since that's an added hypothesis in Rolles theorem. Then you get f ' (c) = 0/(b-a) = 0. That's the conclusion of Rolles theorem.