Rolle's Theorem

From AoPSWiki

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

Statement

Let

Let be continous on and differentiable on

Let

Then such that

Proof

The result is trivial for the case . Hence, let us assume that is a non-constant function.

Let and Without loss of generality, we can assume that

By the Maximum-minimum theorem, such that

Assume if possible

Let

Hence, such that

i.e. ,

Thus we have that if , contradicting the assumption that is a maximum.

Similarly we can show that leads to contradiction.

Therefore,

QED

See Also

• Lagrange's mean value theorem
• Calculus