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 Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.9 Concavity, Inflection Points and Second DerivativesWatch the full walkthrough 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 $convex$ on an interval where $f^{''} (x) \geq 0$ (curve bends upward) and $concave$ where $f^{''} (x) \leq 0$ (curve bends downward).

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

Sign of the second derivative.

Points of inflection

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

y^{''} = 6 x

6 x = 0 \Rightarrow x = 0

, and

y^{''}

changes sign there.

AnswerInflection at

(0, 0)

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

Formula recap

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

Bends upward.

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

Bends downward.

f^{''} (x) = 0 and sign change \Rightarrow 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).

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