One curve. Read it left to right and it tells a whole story — where it climbs, where it turns, where it bends — and f, f′, f″ are just three ways of narrating the same journey. Drag the probe, or press play: the tangent’s tilt is f′, the glowing arc’s cup-or-cap is f″, and every turn plants a flag. By the end you’ve built the whole analysis — the three panels below narrate the same instant in all three dialects.
Each lane is the slope graph of the one above it. That single idea generates every rule you need.
Increasing ⇔ f′>0. Decreasing ⇔ f′<0. A local max/min of f sits where f′=0 and changes sign.
Concave up ⇔ f″>0 (f′ rising). Concave down ⇔ f″<0. An inflection of f sits where f″=0 and changes sign.
f″ is the slope of f′, so a peak or valley of f′ is a zero of f″ — and that’s exactly an inflection of f. The lanes chain together.
Given f′ and f″: f rises where f′>0, bends up where f″>0, has extrema at f′’s zeros and inflections at f″’s zeros. Walk left to right and the shape appears.
Now move it yourself. Grab a handle on f and reshape it — pull up a hill, dig a valley — and f′ and f″ redraw live beneath. The dashed lines keep every turn of f tied to a zero below. Then take a challenge.
Where students most often cross the wires when reading one lane from another:
If f′>0 but f′ is falling, f is still increasing — just concave down (rising, but slowing). A decreasing f′ does not mean a decreasing f.
Where f′ reaches a local maximum, f″=0 — so f has an inflection point there, not a maximum. A max of f is where f′ equals zero, not where f′ peaks.
No — if f′ is still positive, f keeps rising. A falling f′ only means the climb is slowing (concave down).
A peak of f′ is where \(f''=0\): an inflection of f. A max of f is where \(f'=0\) and switches \(+\to-\).
Concavity is the sign of f″ (is f′ rising or falling?), not the sign of f′ itself.
\(f'=0\) or \(f''=0\) are only candidates. A peak, valley, or inflection needs the sign to actually flip.