A Level Maths: Concave and Convex Curves and Points of Inflection Topic Summary and Resources
Video Lessons
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Revision Notes
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Exam Questions
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Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Differentiation — needed throughout for first and second derivatives
- Quadratic Equations — solving f''(x) = 0 often gives quadratics
- Functions — understanding domains is essential
A Level Maths concave and convex curves and points of inflection is the topic where you use the second derivative to understand the shape of a curve in more detail. This builds on Differentiation and provides the rigorous tools for classifying stationary points and identifying inflection points.
A curve is convex (or concave up) on an interval where the second derivative $f''(x) \ge 0$ — visually, the curve curves upwards. It is concave (or concave down) where $f''(x) \le 0$ — curving downwards. A point of inflection is where the curve changes between convex and concave; equivalently, where the second derivative changes sign. You use $f''(x) > 0$ at a stationary point to identify it as a minimum, $f''(x) < 0$ to identify a maximum, and $f''(x) = 0$ needs further investigation — you check whether $f''$ changes sign or use the first derivative test. You sketch curves showing their concavity properties, and apply this analysis to optimisation problems and curve-sketching questions.
Concave and convex curves and points of inflection is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: $f''(x) = 0$ alone does NOT prove a point of inflection — $f''$ must CHANGE SIGN there; a stationary point of inflection (where $f'(x) = 0$ and $f''(x) = 0$) is rarer than the non-stationary kind, where $f'(x)$ is non-zero but $f''(x) = 0$ and changes sign; concave and convex are sometimes called concave-down and concave-up — terminology varies between textbooks; and the second derivative test is INCONCLUSIVE when $f''(x) = 0$ at a stationary point — use the first-derivative sign test instead.