# Rolle's theorem

From Calculus

## Statement

Suppose is a function defined on a closed interval (with ) satisfying the following three conditions:

- is a continuous function on the closed interval . In particular, is (two-sided) continuous at every point in the open interval , right continuous at , and left continuous at .
- is differentiable on the open interval , i.e., the derivative of exists at all points in the open interval .
- .

Then, there exists in the open interval such that .

## Related facts

### Applications

- Lagrange mean value theorem
- Bound relating number of zeros of function and number of zeros of its derivative

## Facts used

## Proof

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | If is zero on all of , then for any choice of | obvious | |||

2 | must attain both its maximum and its minimum values on | Fact (1) | is continuous on | Given-fact combination direct | |

3 | If is not zero on all of , either its absolute maximum value on is positive and attained at a point in the open interval or its absolute minimum value on is negative and attained at a point in the open interval (or possibly both). | Step (2) | [SHOW MORE] | ||

4 | If is a point in at which attains its maximum value or its minimum value, then . |
Fact (2) | is differentiable on | [SHOW MORE] | |

5 | There is a point at which . | Steps (1), (3), (4) | [SHOW MORE] |