A curve is concave up where it cups upward (like a smile) and concave down where it cups downward (like a frown).
Drag the point along the curve
y = x·x·x - 3·xat x = 1: slope ≈ 0
Concavity test
Concave up = slopes increasing as you move right; concave down = slopes decreasing.
Your turn
On what interval is f(x) = x³ − 3x² concave up?
Watch out
Concavity is about f″, not f′. A function can be increasing (f′ > 0) while concave down (f″ < 0) — think of √x growing but flattening.
Test
- f''(x) > 0 — concave up.
- f''(x) < 0 — concave down.
- f''(x) = 0 — possible inflection point.