9.9PureStretch

Concavity and Second Derivatives

The second derivative measures how the gradient itself is changing. Its sign tells you whether a curve is concave or convex, and where it has points of inflection.

26 min Differentiation

Edexcel A level Maths: 9.9 Concavity, Inflection Points and Second DerivativesCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Interpret the sign of the second derivative
Define concave and convex
Find points of inflection
Classify stationary points

Concave and convex

A function is $\textbf{convex}$ on an interval where $f''(x) \ge 0$ (curve bends upward) and $\textbf{concave}$ where $f''(x) \le 0$ (curve bends downward).

f''(x) > 0 \Rightarrow \text{convex}, \quad f''(x) < 0 \Rightarrow \text{concave}

Sign of the second derivative.

Points of inflection

A point of inflection is where the curve changes between concave and convex — so $f''(x) = 0$ $\textbf{and}$ $f''$ changes sign there. (f′′ = 0 alone is not enough.)

Example 1Find the point of inflection of

y = x^3

y'' = 6x

6x = 0 \Rightarrow x = 0

, and

y''

changes sign there.

Answer

Inflection at

(0,0)

Tip — Confirm an inflection by checking f′′ changes sign — not just that f′′ = 0.

Formula recap

f''(x) > 0 \Rightarrow \text{convex}

Bends upward.

f''(x) < 0 \Rightarrow \text{concave}

Bends downward.

f''(x)=0 \text{ and sign change} \Rightarrow \text{inflection}

Point of inflection.

Common mistakes to avoid

Calling any point where f′′ = 0 an inflection.

f′′ must also change sign there.

Mixing up concave and convex.

Convex: f′′ > 0 (smile); concave: f′′ < 0 (frown).

Key takeaways

f′′ > 0: convex; f′′ < 0: concave.
Inflection: f′′ = 0 and f′′ changes sign.
Use f′′ to classify stationary points (min if f′′>0, max if f′′<0).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A curve is convex where:
- $f''(x) > 0$
- $f''(x) < 0$
- $f'(x) = 0$
- $f(x) > 0$
Show answer and explanation
Answer: $f''(x) > 0$
Bends upward.
A point of inflection requires f′′ = 0 and:
- f′′ changes sign
- f = 0
- f′ = 0
- nothing more
Show answer and explanation
Answer: f′′ changes sign
Sign change is essential.
For $y = x^3$ , the inflection point is:
- $(0,0)$
- $(1,1)$
- none
- $(0,1)$
Show answer and explanation
Answer: $(0,0)$
y′′ = 6x changes sign at 0.
If f′′(x) < 0 at a stationary point, it is a:
- maximum
- minimum
- inflection
- cusp
Show answer and explanation
Answer: maximum
Concave down.
A curve is concave where:
- $f''(x) < 0$
- $f''(x) > 0$
- $f'(x) > 0$
- $f(x) < 0$
Show answer and explanation
Answer: $f''(x) < 0$
Bends downward.

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