The mean value theorem is still valid in a slightly more general setting.

It can be generalized to more variables according by the mean value theorem for divided differences.

It is reminiscent of the intermediate value theorem for continuous functions.

If the degree is odd, then by the intermediate value theorem at least one of the roots will be real.

Then by the extreme value theorem, attains its global minimum.

It can be obtained from the mean value theorem by choosing .

There will always be a point at which this happens, because of the intermediate value theorem.

From the intermediate value theorem, has at least n roots.

Indeed, it is needed to prove both the mean value theorem and the existence of Taylor series.

That fact can also be proven by using the intermediate value theorem.