Concept preview

What you will be able to do

1. 01Read across the three graphs $f$, $f'$, and $f''$ fluently: translate “$f$ increasing” into “$f'>0$”, “$f$ concave up” into “$f''>0$ (equivalently $f'$ increasing)”, and locate local extrema and inflection points on whichever graph is in front of you.
2. 02Reconstruct qualitative facts about $f$ from a graph or table of $f'$ alone (or of $f''$ alone), using must-be-true reasoning rather than guessing at the shape of $f$.
3. 03Diagnose and avoid the single most common error in this topic — mistaking the graph of $f'$ for the graph of $f$ — by naming, for every conclusion, which derivative controls it.

Begin with the first derivative. The value of $f'(x)$ is the slope of $f$ at $x$, so its SIGN is a verdict on direction: where $f'>0$ the graph of $f$ is climbing, where $f'<0$ it is descending, and where $f'=0$ the tangent is momentarily level. Because a smooth graph can only switch from rising to falling by passing through a level instant, the candidates for local maxima and minima of $f$ are exactly the $x$-values where $f'$ crosses zero. But crossing zero is not enough by itself — you must check HOW $f'$ crosses. If $f'$ goes from positive to negative, $f$ was rising then falling, so that point is a local maximum. If $f'$ goes from negative to positive, $f$ was falling then rising, so it is a local minimum. If $f'$ merely touches zero without changing sign, $f$ has a flat spot but no extremum there.