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Rolle's Theorem

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Rolle's theorem is an important theorem among the class of results regarding the value of the derivative on an interval.

Statement

Let f:[a,b]\rightarrow\mathbb{R}

Let f be continous on [a,b] and differentiable on (a,b)

Let f(a)=f(b)

Then \exists c\in (a,b) such that f'(c)=0

Proof

The result is trivial for the case f([a,b])=\{f(a)\}. Hence, let us assume that f is a non-constant function.

Let M=\sup\{f([a,b])\} and m=\inf\{f([a,b])\} Without loss of generality, we can assume that M\neq f(a)

By the Maximum-minimum theorem, \exists c\in (a,b) such that f(c)=M

Assume if possible f'(c)>0

Let \epsilon=\frac{f'(c)}{2}

Hence, \exists \delta>0 such that x\in V_{\delta}(c)\implies |\frac{f(x)-f(c)}{x-c}-f'(c)|<\epsilon

i.e. \forall x\in V_{\delta}(c), \frac{f(x)-f(c)}{x-c}>0

Thus we have that f(x)>f(c) if x\in (c,c+\delta), contradicting the assumption that f(c) is a maximum.

Similarly we can show that f'(c)<0 leads to contradiction.

Therefore, f'(c)=0

QED

See Also

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