Inflection points are where the curve changes from concave up to concave down (or vice versa) — second derivative changes sign.
Drag the point along the curve
y = x·x·x - 3·xat x = 1: slope ≈ 0
Inflection point
f″(c) = 0 alone isn't enough — the concavity must actually flip. For y = x⁴, f″(0) = 0 but it stays concave up.
Your turn
Find the inflection point of f(x) = x³ − 6x² + 9x.
At an inflection point the curve crosses its own tangent line — concavity swaps from cupping up to cupping down (or vice versa).
Try it
f(x) = x³, find inflection point
f''(x) = 6x. Sign changes at x = 0. Inflection at (0, 0).