AP® Calculus AB · Unit 5 · Topic 5.6
Points of Inflection
A point of inflection is where a curve's concavity actually switches — from cupping up to cupping down or back — which happens only where f'' changes sign.
A curve concave up looks like a cup; concave down looks like a frown. To go from one to the other, the curve can't jump — it has to pass through a moment where the bending stops, reverses, and starts again the other way. That moment is a point of inflection. Since f'' measures concavity (positive means concave up, negative means concave down), a real switch in shape requires f'' to actually cross zero, not just touch it.
This is why you always check the sign of f'' on both sides of a candidate point, not just whether f''=0 there. Set f''(x)=0, find the x-values, then test values slightly left and right of each one. If the sign of f'' flips, you have a genuine inflection point — plug the x-value back into the original f(x) to get the full coordinate if needed. If the sign stays the same on both sides, the curve touched that concavity but never switched, so it's not an inflection point, just a false alarm from the algebra.
This sign-change idea is the same logic behind the second derivative test for extrema: a local min needs f'' positive (concave up, cupping around a valley), a local max needs f'' negative. Points of inflection and extrema are both read off f'', but they ask different questions — extrema care about the sign of f'' at a single critical point, inflection points care about f'' changing sign across a point. Together they let you sketch a curve's complete shape: where it rises and falls, and where it bends which way.
Key terms
- Point of inflection
- A point where the concavity of a curve changes from up to down or down to up.
- Concavity
- The direction a curve bends, described as concave up (like a cup) or concave down (like a frown).
- Second derivative test
- A method using the sign of f'' at a critical point to classify it as a local max or local min.
- Candidate point
- An x-value where f''=0 or is undefined, which must still be checked for an actual sign change.
Watch it explained
- How it worksHow does a sign change in the second derivative mark a genuine point of inflection?
- Worked exampleFind every point of inflection of f(x) = x5 - 5x4.
- Common mistakeWhy isn't every zero of the second derivative a point of inflection?
- Ties togetherHow does a sign change in the second derivative connect to the second derivative test for classifying extrema?
Check yourself
End of topic.