Using the Mean Value Theorem: Showing One-to-One Behavior

[Swamifez]

Homework Statement

Use the Mean Value Theorem to show that:

a)Suppose f is a diferentiable function on the interval a < b, and suppose f '(x) is not equal to 0 for all x Element Symbol (a,b). Show that f is one-to-one on the interval (a,b).

b) Assume that |f ' (x)| < or equal to C < 1 for all x. Show that f (x) = x has at most one solution.

Homework Equations

The Attempt at a Solution

Suppose f is not one-one on the interval then there exists u, v in (a,b) such that f(u)=f(v).
Then by the mean value theorem there exists a point w in (u,v) such that f'(w)=0, a contradiction.

Don't know where to go after that.

 

[quasar987]

It's great that you found the homework section of the board, but please do not double post. You posted this question in another part of the forum already. If you realize that you've posted in the wrong section of the forum, just let the moderators move the thread to the right section.