Convex, concave and points of inflection

Using the second derivative to decide where a curve is convex (f'' > 0) or concave (f'' < 0), and locating points of inflection where the concavity changes — distinct from a stationary point.

Pure · Differentiation

Convex, concave and points of inflection

Using the second derivative to decide where a curve is convex (f'' > 0) or concave (f'' < 0), and locating points of inflection where the concavity changes — distinct from a stationary point.

Why it works

The first derivative

f'(x)

tells you whether a curve is going up or down. The second derivative

f''(x)

tells you which way it bends — and that shape information is what this section is about.

Where $f''(x) > 0$ the gradient is increasing, so the curve bends upward like a

valley: it is convex (often said "concave up"). It sits above its tangents.

Where $f''(x) < 0$ the gradient is decreasing, so the curve bends downward like

a hill: it is concave (or "concave down"). It sits below its tangents.

A point of inflection is where the curve changes from one to the other — from bending up to bending down or vice versa. Since

f''

is positive on one side and negative on the other, at the inflection itself

f''(x) = 0 \quad\text{and } f'' \text{ changes sign}.

Both conditions matter.

f''(x) = 0

on its own is not enough: for

y = x^4

we get

f''(0) = 0

, but

f'' = 12x^2 \ge 0

on both sides, so the curve stays convex and the origin is a minimum, not an inflection. You must check that $f''$ actually changes sign — the sure test for an inflection.

The simplest example is

y = x^3

. Here

f'' = 6x

, which is negative for

x < 0

(concave) and positive for

x > 0

(convex), changing sign at

x = 0

. So the origin is a point of inflection.Inflection vs stationary point. These are independent ideas and a frequent source of confusion:

A stationary point of inflection also has $f'(x) = 0$ — the curve is

momentarily flat and changing concavity, as at the origin of

y = x^3

. This is the third option (besides max and min) when the second-derivative test fails: if

f'(a) = 0

and

f''(a) = 0

, you cannot conclude a min — check for an inflection.

A non-stationary point of inflection has $f'(x) \ne 0$ — the curve changes

bending while still sloping, like the middle of an S-shaped cubic. Most points of inflection are this kind.

So to find points of inflection: compute

f''(x)

, solve

f''(x) = 0

, and confirm the sign of

f''

flips across each root.