Concept preview

What you will be able to do

1. 01State the First Derivative Test: at a critical number $c$, if $f'$ changes from $+$ to $-$ then $f(c)$ is a local (relative) MAXIMUM, if $f'$ changes from $-$ to $+$ then $f(c)$ is a local MINIMUM, and if $f'$ does NOT change sign then $c$ is neither.
2. 02Justify a classification of a relative extremum with an explicit sign analysis of $f'$ on both sides of the critical number, rather than merely asserting max or min.
3. 03Apply the test to polynomials — finding every critical number, testing the sign of $f'$ just before and just after each, and reporting the location and type of each relative extremum.

Suppose $c$ is a critical number of a continuous function $f$ (so $f'(c)=0$ or $f'(c)$ is undefined). Look at the sign of $f'$ immediately to the LEFT of $c$ and immediately to the RIGHT of $c$. Three things can happen. (1) If $f'$ changes from POSITIVE to NEGATIVE as $x$ increases through $c$, then $f$ rises up to $c$ and falls afterward, so $f$ has a local MAXIMUM at $c$. (2) If $f'$ changes from NEGATIVE to POSITIVE, then $f$ falls into $c$ and rises afterward, so $f$ has a local MINIMUM at $c$. (3) If $f'$ does NOT change sign — it is positive on both sides, or negative on both sides — then $f$ keeps going the same direction and $c$ is NEITHER a max nor a min.