# Other applications of the (first) derivative.

A consequence of Lagrange's theorem (in its second form 8.5) is the following well-known theorem:

Theorem 6.9.1   Let be a function defined and differentiable on an interval .
1. If , , then is constant on .
2. If , , then is strictly increasing on .
3. If , , then is increasing on .
4. If , , then is strictly decreasing on .
5. If , , then is decreasing on .

Example 6.9.2   Take ; as is a polynomial function, it is differentiable on .

.

 or

Thus, is strictly increasing on and on and is strictly decreasing on . It has a maximum at 1 and a minimum at 1 (v.s. Def. 5.6).

Theorem 6.9.3   If is differentiable at and has an extremum at , then .

For example, take . Then has no extremum, but . We will se later that such a point is called an inflection point (see Definition 10.8 and Figure8).

Noah Dana-Picard 2007-12-28