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

- If , , then is constant on .
- If , , then is strictly increasing on .
- If , , then is increasing on .
- If , , then is strictly decreasing on .
- If , , then is decreasing 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).

Noah Dana-Picard 2007-12-28