AP® Precalculus · Unit 1 · Topic 1.4
Points of Inflection
A point of inflection is where a curve switches concavity, from bending upward to bending downward or vice versa. It marks where the rate of change itself stops increasing and starts decreasing, or the reverse.
A curve isn't just going up or down, it's also bending a certain way. Concave up means the curve bends like a cup, and the slope is increasing as you move left to right. Concave down means it bends like a frown, and the slope is decreasing. A point of inflection is the spot where the bending flips from one to the other. The curve doesn't stop increasing or decreasing there necessarily, it just changes how it's curving.
Mechanically, concavity is governed by the second derivative. Where f''(x) > 0 the graph is concave up, where f''(x) < 0 it's concave down. A candidate point of inflection shows up where f''(x) = 0, but that alone doesn't guarantee a real switch, you need the sign of f'' to actually change on either side, or equivalently f'''(x) must be nonzero at that point for a cubic. For f(x) = x3 - 6x2 + 9x, f'(x) = 3x2 - 12x + 9 and f''(x) = 6x - 12, which is zero at x = 2. Since f'''(x) = 6, never zero, concavity really does flip there: f'' is negative for x < 2 (concave down) and positive for x > 2 (concave up).
This sits right next to extrema in the same toolkit of derivative tests, but it answers a different question. Maxima and minima come from where f'(x) = 0 and the slope changes sign. Points of inflection come from where f''(x) changes sign, regardless of what the slope is doing. A cubic like this one has no maximum or minimum that's global, but it does have exactly one inflection point, which is often the most interesting feature of its graph.
Key terms
- Point of inflection
- A point on a graph where concavity switches from concave up to concave down or vice versa.
- Concave up
- The graph bends upward like a cup, and the slope is increasing.
- Concave down
- The graph bends downward like a frown, and the slope is decreasing.
- Second derivative
- The derivative of the derivative, f''(x), which measures how the slope itself is changing.
- Third derivative test
- If f''(x) = 0 and f'''(x) is nonzero at that point, concavity actually changes there, confirming a true inflection point.
Watch it explained
- How it worksHow does a point of inflection mark the moment the rate of change stops growing and starts shrinking?
- Worked exampleFind the point of inflection of f(x) = x³ - 6x² + 9x and show the bend flipping right there.
- Common mistakeWhy isn't a point of inflection the same thing as a maximum or a minimum?
Check yourself
End of topic.