Extrema using First Derivative Test

Once the intervals on which a function is increasing or decreasing are known, locating its relative extrema is straightforward using the following test.

Theorem 7.10 (First Derivative Test). Let $\big(c,f(c)\big)$ be a critical point of $f(x)$, with $f$ continuous on an open interval $I$ containing $c$ and differentiable on $I$ except possibly at $c$. Moving across $I$ from left to right:

1. If $f'(x)$ changes from negative to positive at $c$, then $f(x)$ has a local minimum $f(c)$.
2. If $f'(x)$ changes from positive to negative at $c$, then $f(x)$ has a local maximum $f(c)$.
3. If $f'(x)$ is positive on both sides of $c$, or negative on both sides of $c$, then $f(c)$ is neither a local minimum nor a local maximum. Working procedure (used throughout Exercise 7.6 Q2 and, later, Exercise 7.7 Q3):

1. Find the critical numbers (solve $f'(x)=0$; note where $f'$ is undefined).
2. Tabulate the sign of $f'(x)$ on each interval between consecutive critical numbers (this simultaneously reads off the monotonicity — Theorem 7.7 — and feeds directly into the First Derivative Test).
3. Classify each critical number using the sign-change pattern above, and compute the extreme value $f(c)$ itself. …