## Quick Reference

The following result concerning the existence of stationary points of a function *f*:

Theorem

Let *f* be a function which is continuous on [*a*, *b*] and differentiable in (*a*, *b*), such that *f*(*a*)=*f*(*b*). (Some authors require that *f*(*a*)=*f*(*b*)=0.) Then there is a number *c* with *a* < *c* < *b* such that *f′*(*c*)=0. The result stated in the theorem can be expressed as a statement about the graph of *f*: with appropriate conditions on *f*, between any two points on the graph *y*=*f*(*x*) that are level with each other, there must be a stationary point; that is, a point at which the tangent is horizontal. The theorem is, in fact, a special case of the Mean Value Theorem; however, it is normal to establish Rolle's Theorem first and deduce the Mean Value Theorem from it. A rigorous proof relies on the non-elementary result that a continuous function on a closed interval attains its bounds.

*Subjects:*
Mathematics.

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