Theorem 1   If the real function is

1. continuous on the closed interval and
2. differentiable on the open interval ,

then the function has on the interval a least value and a greatest value. These are always got in the end of the interval or in the zero of the derivative.

Remark 1. If the preconditions of the theorem are fulfilled by a function , then one needs only to determine the values of in the end points and of the interval and in the zeros of the derivative inside the interval; then the least and the greatest value are found among those values.

Remark 2. Note that the theorem does not require anything of the derivative in the points and ; one needs not even the right-sided derivative in or the left-sided derivative in . Thus e.g. the function , fulfilling the conditions of the theorem on the interval but not having such one-sided derivatives, gains its least value in the end-point and its greatest value in the zero of the derivative.