**Theorem CCD.MVT. The Mean Value Theorem:**

If $f$ is a continuous function on the interval
$[a,b]$ and a differentiable function on the interval $(a,b)$, then there is a number $c \in (a,b)$
where $$f'(c) = \frac{f(b) - f(a)}{b-a}$$ or $$f(b) = f(a) + f'(c) (b-a) $$ .

CCD.MVT.FDA : First Derivative Analysis. Visualizing the derivative for an interval with the "derivative vector" in a mapping diagram supports first derivative analysis for monotonic function behavior.

If $f''(x) \gt 0$ for an interval then $f'(x)$ is increasing for that interval and $f(x)$ is accelerating for that interval.