Inflection points
Also known as: inflection point, point of inflection
An inflection point is a point on a curve where the concavity changes — the graph switches from curving upward to curving downward, or the reverse. It occurs where the second derivative changes sign.
Concavity describes which way a curve bends. A function is concave up where its graph opens upward like a bowl, and concave down where it opens downward like a dome. An inflection point is exactly where that behavior flips. At such a point the curve stops bending one way and begins bending the other, even though the function itself may still be increasing or decreasing throughout.
Because concavity is governed by the second derivative, you find candidates by solving f''(x) = 0 or locating where f''(x) fails to exist. Those are only candidates: an inflection point requires f'' to actually change sign there. For f(x) = x³, f''(x) = 6x is zero at x = 0 and changes from negative to positive, so (0, 0) is an inflection point. For f(x) = x⁴, f''(x) = 12x² is also zero at x = 0, but it stays positive on both sides — the curve is concave up throughout and there is no inflection point.
Inflection points carry real meaning beyond the graph. On a position curve they mark where acceleration changes sign; on a growth curve they mark the moment the rate of growth stops accelerating and starts slowing. The normal distribution has inflection points located exactly one standard deviation on either side of the mean, which is why the bell curve visibly straightens out at those points. A titration curve has an inflection point at its equivalence point, where pH changes fastest.
AP Calculus AB tests inflection points directly in the analytical applications of derivatives, including questions that give you a graph of f' or f'' and ask where f has an inflection point. AP Statistics uses them descriptively when characterizing the normal curve, and the MCAT applies the same idea to titration curves in general chemistry.
Key takeaways
- An inflection point is where a curve changes concavity, from concave up to concave down or vice versa.
- Candidates occur where f''(x) = 0 or f''(x) is undefined, but the second derivative must actually change sign.
- f(x) = x³ has an inflection point at x = 0; f(x) = x⁴ does not, even though f''(0) = 0 for both.
- The normal distribution's inflection points sit one standard deviation above and below the mean.