Concavity and inflection points

Concave up vs concave down

Test for concavity. If f is twice differentiable on an interval I, then f is concave up on I if f″(x) > 0 for all x in I, and concave down on I if f″(x) < 0 for all x in I.

What is a point of inflection?

If f is continuous at a and changes concavity at a, the point (a, f(a)) is an inflection point. To find candidates, look where f″(x) = 0 or f″ is undefined, then check that the sign of f″ actually changes there.

f″ = 0 is not enough. For f(x) = x⁴, f″(x) = 12x² is 0 at x = 0, but it's positive on both sides, so the graph is concave up throughout and there's no inflection point.

Example 1: f(x) = x³ − 6x² + 9x + 30

1. First derivative: f′(x) = 3x² − 12x + 9.
2. Second derivative: f″(x) = 6x − 12.
3. Solve f″(x) = 0: x = 2.
4. Test each side. At x = 0, f″(0) = −12 < 0. At x = 3, f″(3) = 6 > 0.
5. So f is concave down on (−∞, 2) and concave up on (2, ∞).
6. The concavity changes at x = 2, and f(2) = 8 − 24 + 18 + 30 = 32, so the inflection point is (2, 32).

The second derivative test

Suppose f′(c) = 0 and f″ is continuous near c.

• If f″(c) > 0, f has a local minimum at c (concave up, like the bottom of a cup).
• If f″(c) < 0, f has a local maximum at c.
• If f″(c) = 0, the test is inconclusive. Use the first derivative test instead. For example, x⁴ has a minimum at 0, −x⁴ has a maximum, and x³ has neither, yet all three have f″(0) = 0.

Example 2: f(x) = x⁵ − 5x³

1. f′(x) = 5x⁴ − 15x² = 5x²(x² − 3). Critical points: x = 0 and x = ±√3.
2. f″(x) = 20x³ − 30x = 10x(2x² − 3).
3. At x = −√3: f″ = −30√3 < 0, so a local maximum.
4. At x = √3: f″ = 30√3 > 0, so a local minimum.
5. At x = 0: f″(0) = 0, inconclusive. f′ is negative on both sides of 0 (5x² ≥ 0 and x² − 3 < 0 near 0), so f is decreasing through 0: no local extremum.
6. Inflection points: f″ = 0 at x = 0 and x = ±√(3/2), and f″ changes sign at each, so all three are inflection points.

Reading concavity in the real world

If f is a position, f″ is acceleration: concave up means speeding up in the positive direction. On a growth curve, an inflection point is where growth stops accelerating and starts slowing down.

Find inflection points with the inflection point calculator, compute f′ and f″ with the derivative calculator, and see the shape on the graphing calculator. For the first-derivative side of the picture, see the mean value theorem.

Sources: OpenStax Calculus Volume 1, 4.5 Derivatives and the Shape of a Graph.

Every worked example on this page was checked with Calcist's own engine and an independent reference.