Monotonicity of convex function

1. Apr 17, 2010

hellbike

$$f:(a,\infty)->R$$
i want to prove, that, if function is convex, then:

if exist $$x_1 \in R$$, exist $$x_2>x_1$$ : $$f(x_2)>f(x_1)$$
then:
for all $$x_3>x_2$$ for all$$x_4>x_3$$ : $$f(x_4)\ge f(x_3)\ge f(x_2)$$

in other words:
convex function is either decreasing on whole domain, or it starts to increase from some point and then is increasing from that point to the end of domain

We dont assume that function is differential.

With assumption that $$f(x_2)>f(x_1)$$ i should get something from definition of convex function, but i don't know how to do it.

Last edited: Apr 17, 2010
2. Apr 17, 2010

g_edgar

First show that convex implies this: If $a < b < c$, then
$$\frac{f(b)-f(a)}{b-a} \le \frac{f(c)-f(b)}{c-b}$$