I am struggling with this algebraic development and wonder if any of you can help me out. Would be very nice if you please could explain your "strategy" when doing so as well.
First of all:
$\xi = (1 - \theta)x_1 + \theta x_2$
I cannot understand how the right side is equal to the left side.
$(1- \theta) f(x_1) + \theta f(x_2) - f((1-\theta)x_1 + \theta x_2)$ $ =$ $(1-\theta)[f(x_1-f(\xi)] + \theta[f(x_2)-f(\xi)]$
This is part of a proof of convex functions and the proof has more steps beyond this which will incorporate the mean value theorem. I think I can manage the next steps if I get some help with this first one :)
Thank you very much!