AP Calculus AB · Summary

Unit 5 summary

1. Mean Value Theorem: continuous on $[a, b]$ and differentiable on $(a, b)$ means $f'(c)$ equals the average rate for some $c$ in $(a, b)$.
2. Extreme Value Theorem: a function continuous on a closed interval has an absolute maximum and an absolute minimum there.
3. Critical points are interior inputs where $f' = 0$ or $f'$ is undefined. A relative extremum needs a sign change of $f'$.
4. For absolute extrema on a closed interval, compare f at every critical point and both endpoints.
5. $f' > 0$ means increasing and $f' < 0$ decreasing; $f'' > 0$ means concave up and $f'' < 0$ concave down. An inflection point needs $f''$ to change sign.
6. Second derivative test: $f'(c) = 0$ with $f''(c) < 0$ is a relative maximum; with $f''(c) > 0$, a relative minimum.
7. In optimization, write the objective function in one variable on its feasible domain, find its critical points, and justify the absolute extremum.