## Quick Reference

The following theorem, which has very important consequences in differential calculus:

Theorem

Let *f* be a function that is continuous on [*a*, *b*] and differentiable in (*a*, *b*). Then there is a number *c* with *a* < *c* < *b* such that

The result stated in the theorem can be expressed as a statement about the graph of *f*: if *A*, with coordinates (*a*, *f*(*a*), and *B*, with coordinates (*b*, *f*(*b*), are the points on the graph corresponding to the end-points of the interval, there must be a point *C* on the graph between *A* and *B* at which the tangent is parallel to the chord *AB*.

The theorem is normally deduced from Rolle's Theorem, which is in fact the special case of the Mean Value Theorem in which *f*(*a*)=*f*(*b*). A rigorous proof of either theorem relies on the non-elementary result that a continuous function on a closed interval attains its bounds. The Mean Value Theorem has immediate corollaries, such as the following. With the appropriate conditions on *f*,*f′*(*x*)=0 for all *x*, then *f* is a constant function,*f′*(*x*)>0 for all *x*, then *f* is strictly increasing.The important Taylor's Theorem can also be seen as an extension of the Mean Value Theorem.

*f′*(*x*)=0 for all *x*, then *f* is a constant function,

*f′*(*x*)>0 for all *x*, then *f* is strictly increasing.

*Subjects:*
Mathematics.

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