MATH 140 Lecture Notes - Mean Value Theorem
14 Oct 2022
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Rolle"s theorem: f continuous on [a, b] and di erentiable on (a, b). Then there is some c in (a, b) such that f (c) = 0. Since f is continuous on [a, b], by the maximum-minimum theorem, f has a maximum and a minimum value on [a, b]. Suppose rst that both the max and the min happen at the endpoints. But since f (a) = f (b), this means that the max equals the min so f is constant. Hence f (c) = 0 for all c in (a, b). Otherwise, there is at least one extreme value at some number c in (a, b). Let f be continuous on [a, b] and di erentiable on (a, b). Then there is some c in (a, b) such that f (c) = Let g(x) = f (x) (x a) f (a) + f (b) f (a) Then g is continuous on [a, b] and di erentibale on (a, b).
and 
for
. Prove that
. [Hint:
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