Point of inflection

Definition

A point of inflection is where a curve changes from concave up to concave down (or the reverse). It is found by solving f″(x) = 0. For a cubic, it lies exactly halfway between the two stationary points.

Worked example

For f(x) = x³ − 6x², f″(x) = 6x − 12 = 0 gives x = 2.

Where it appears in the exam

Paper 1 cubic graph questions ask for the point of inflection or for where the graph is concave up or concave down.

Step-by-step method

1. Find f″(x).
2. Set f″(x) = 0 and solve for x.
3. Substitute into f(x) to find the y-coordinate.
4. Use f″(x) > 0 for concave up and f″(x) < 0 for concave down.

Common mistake

Confusing it with a stationary point: f′(x) does not have to be zero at a point of inflection.

Related calculus terms

• First principles
• Stationary point
• Optimisation

Frequently asked questions

What does point of inflection mean in Grade 12 Maths?

A point of inflection is where a curve changes from concave up to concave down (or the reverse). It is found by solving f″(x) = 0. For a cubic, it lies exactly halfway between the two stationary points.

Which NSC paper tests point of inflection?

Point of inflection belongs to the Calculus section of CAPS Grade 12 Mathematics, which is examined in Paper 1.

How do I practise questions involving point of inflection?

Snap a question that uses point of inflection into the Snap&Learn AI solver, or work through the calculus topic guide and NSC past papers.