Concavity and points of inflection

• The curve is concave up (∪, "holds water") where f″(x) > 0 and concave down (∩) where f″(x) < 0.
• A point of inflection is where the concavity changes: solve f″(x) = 0 and check that the sign of f″ really changes across the root.
• At a point of inflection the tangent crosses the curve, and the value of f′ there is a maximum or a minimum (the slope is extreme).
• For a cubic, the point of inflection is the midpoint of the segment joining the maximum and the minimum; its x-coordinate is the average of the two critical numbers.

Worked example

f(x) = x³ − 6x² + 9x + 2 ⟹ f″ = 6x − 12 = 0 ⟹ x = 2, f(2) = 4. Sign of f″: negative before 2, positive after ⟹ point of inflection (2, 4) — indeed the midpoint of (1, 6) and (3, 2).

When there is no inflection point

f(x) = x⁴ has f″ = 12x², which is zero at x = 0 but positive on both sides, so x = 0 is a minimum, not an inflection point. Zero second derivative alone is never enough.