Concavity and inflection points
What you'll learn
Use the second derivative to tell which way a curve bends — and where it changes its bend.
The first derivative says which way a curve is going. The second derivative says which way it's bending — and bending is what separates a gentle approach from a sharp one.
Which way does it bend?
The second derivative f″ is the derivative of f′ — the rate at which the slope itself changes:
- f″(x) > 0 → the slope is increasing → the curve is concave up (it holds water, ∪).
- f″(x) < 0 → the slope is decreasing → the curve is concave down (it spills, ∩).
- f″(x) = 0 and the concavity flips → an inflection point.
Reading a peak or valley faster
Concavity gives a shortcut for classifying critical points — the second- derivative test. At a critical point where f′ = 0:
| f″ at that point | the point is a |
|---|---|
| f″ > 0 (concave up) | local minimum |
| f″ < 0 (concave down) | local maximum |
| f″ = 0 | inconclusive — fall back to the first-derivative test |
A valley curves upward, a peak curves downward — so the sign of f″ names it in one step.
Worked example
For f(x) = x³: f′(x) = 3x² and f″(x) = 6x. Then f″ < 0 for x < 0 (concave down) and f″ > 0 for x > 0 (concave up), so x = 0 is an inflection point — the curve switches its bend there, even though it never stops rising.
Why this matters
Concavity is the difference between accelerating and decelerating growth. An epidemic curve bending down (f″ < 0) — still rising, but slowing — is the first sign the peak is coming. The inflection point is exactly where that turn begins.
Check your understanding
Question 1 of 2
If f″(x) > 0, the graph of f is: