Proving an Inequality for a Differentiable Function

Question:

Hello everyone,

I'm currently studying real analysis and I've come across a problem that I'm having trouble with. The problem is as follows:

Suppose that $$g$$ is differentiable on $$(a,b)$$. Show that the inequality

$$\frac{g(x_0)-g(x_1)}{x_0-x_1}\le\frac{g(x_2)-g(x_0)}{x_2-x_0}$$

holds for all $$x_1,x_0,x_2$$ such that $$x_1<x_0<x_2$$ if and only if the function $$g'$$ is increasing.

My Attempts:

I've tried to apply the Mean Value Theorem since the conditions of the theorem seem to be satisfied here. However, I'm not sure how to proceed from there.

My Background:

I have a basic understanding of calculus and real analysis, including concepts like limits, continuity, and differentiability. I'm familiar with the Mean Value Theorem and its applications, but I'm having trouble applying it in this context.

Context:

This problem is from a set of practice problems that I'm working on to improve my understanding of real analysis. I believe understanding the solution to this problem will help me better understand the properties of differentiable functions and how to work with inequalities involving these functions.