So, I'm reading an "easy" proof of the Central Limit Theorem in the book Probability and Mathematical Statistics, by Sahoo, and there is a point where the moment generating function is expanded in a Taylor series about $t=0$
$$M(t) =M(0) + M'(0)t + \frac{1}{2} M''(\eta)t^2$$ with $\eta\in(0,t)$
So my question, where did that $\eta$ come from?
It's basically saying that $$\frac{1}{2} M''(\eta)t^2 = \frac{1}{2} M''(0)t^2 + \mathcal{O}(3)$$
It has the looks of a mean value theorem to me, but I cannot seem to connect the two.
Also, I didn't know how to better phrase the question. I suspect that if I did I could have pulled the correct theorem to breach the gap