The curve is f; the graph stacked beneath it is f′. Drag the point — its tangent tilts, glowing green where the slope is positive and red where it’s negative — while a marker rides f′ below. Every peak and valley of f sits exactly where f′ crosses zero. Press Run the test and it sweeps across, reads the sign of f′ on each side of every critical point, and stamps the verdict: +→− a max, −→+ a min, no flip → a shelf.
It looks like a rule you just memorize — but it falls straight out of last lesson’s Mean Value Theorem. Four steps take you from “positive slope” to the full First-Derivative Test.
Everything you need to classify a peak or valley lives in the sign of f′ on either side of a critical point.
Find the critical points — where f′=0 or f′ DNE. At each one, check the sign of f′ just to its left and just to its right:
No sign change ⇒ not an extremum — just a level shelf.
Mark every critical point on a number line; they cut it into intervals. Pick one test value inside each, plug into f′, and record a + or −.
The pattern of signs names every increasing/decreasing interval and flags every extremum at once — exactly what the f′ lane traces beneath the curve above.
A critical point is only a candidate. Two classic traps — one where f′=0 but nothing turns, one where f′ doesn’t even exist yet there’s a genuine minimum:
For f(x)=x³, f′(x)=3x² is zero at x=0 — but f′ is positive on both sides. The sign never changes (+ → +), so the graph just keeps climbing. Critical, but not an extremum.
For f(x)=|x|, f′ is −1 then +1 — undefined right at x=0. Yet the sign flips − → +, so x=0 is a true minimum. Never skip the points where f′ fails to exist.
On an interval where f′>0, f is increasing; where f′<0, it is decreasing. State intervals in terms of x.
Interior points where f′(x)=0 or f′(x) does not exist. Every local extremum is a critical point — but not every critical point is an extremum.
+ → − means the graph rose then fell — a local maximum. − → + means it fell then rose — a local minimum. Match the arrows to the picture.
The first-derivative test finds local (relative) extrema — biggest/smallest nearby. For the absolute extremum on a closed interval, still compare those against the endpoints.
Increase, decrease, and extrema are decided by the sign of the derivative — not by whether f itself is positive or negative.
You need a sign change. \(f(x)=x^3\) has \(f'(0)=0\) but keeps rising — a shelf, not a peak.
Corners and cusps — like \(|x|\) or \(x^{2/3}\) at 0 — are critical points too, and can be real extrema.
\(+\to-\) is a max (up then down); \(-\to+\) is a min (down then up). Keep the order straight.