## Mathematical Analysis## Differential Calculus |

## What is the Mean-Value Theorem? |

The Mean-Value Theorem has nothing to do with crankiness, but it *is* one of the most important theoretical tools in the calculus. In written terms, it is defined as the following: If *f* (x) is defined and continuous on the interval [a, b], and differentiable on *(a*, b), then there is at least one number on the interval *(a, b)*—or *a < c=""><>*—such that:

When *f(a) =* *f*(*b*), this is a special case called *Rolle’s Theorem*, when we know that *f* (*c*) will equal zero. Interpreting this, we know that there is a point on (*a*, *b*) that has a horizontal tangent.

It is also true that the Mean-Value Theorem can be put in terms of slopes. Thus, the last part of the above equation (on the right of the equal sign) represents the slope of a line passing through (*a*, *f*(*a*)) and *(b*, *f*(*b*)). Thus, this theory states that there is a point *c* ∈ (*a*, *b*), such that the tangent line is parallel to a line passing through the two points.