Proving f is strictly increasing when f' > 0 except at one point

[Seb97]

Homework Statement

If f is differentiable in an interval I and f' >0 throughout I, except possibly at a single point where f' >=0 then f is stictly incresing on I

Homework Equations

The Attempt at a Solution

Ok what I have is I let f'(x) >0. I let a and b two points in the interval with a<b. then for some x in (a,b) with
F'x= (f(b)-f(a))/b-a
but f'(x)>0 for all x in (a,b) so
(f(b)-f(a))/(b-a) >0

since b-a>0 it follows that f(b)>f(a)


What you can see I have proved that it is incresing in the interval but I am not sure what to do when f'=0 any help would be much appreciated as I have been told that it is not fully correct.

 

[ystael]

Do you have the integral and the fundamental theorem of calculus available to you, or only the derivative?

 

[Seb97]

Ah its cool. I managed to work it out. And I am pretty convinced that it works.
Thanks anyway appreciate it.