Points of inflection

A point of inflection is a point on a curve where it changes from bending one way to bending the other: from convex to concave, or from concave to convex. At a point of inflection the second derivative, f″(x), is zero and changes sign. If the gradient is also zero there, it is a stationary point of inflection; if not, it is a non-stationary point of inflection.

What is a point of inflection?

Picture driving along a curve from left to right. On a convex section you turn left: the curve bends upwards like a bowl and its gradient is increasing, so f″(x) > 0. On a concave section you turn right: the curve bends downwards like an arch and its gradient is decreasing, so f″(x) < 0. A point of inflection is where you switch from one to the other.

In the US these are called inflection points, and convex and concave are called concave up and concave down. Older British books spell it point of inflexion. They all mean the same thing.

How to find a point of inflection

1. Differentiate twice to find f′(x) and f″(x).
2. Solve f″(x) = 0. These x values are the possible points of inflection.
3. Check that f″(x) changes sign by working it out just either side of each x value. If it changes sign, it is a point of inflection. If not, it isn't.
4. Find the y coordinate by substituting x into the original equation.
5. Decide which kind it is: work out f′(x) there. Zero means a stationary point of inflection; anything else means non-stationary.

Example 1: a cubic

Find the point of inflection on the curve y = x³ + 3x² − 24x + 2.

Set the second derivative to zero: 6x + 6 = 0, so x = −1. Check the sign either side: at x = −2 it is −6, and at x = 0 it is 6. It changes from negative to positive, so the curve changes from concave to convex. The y coordinate is (−1)³ + 3(−1)² − 24(−1) + 2 = 28.

The gradient there is 3 − 6 − 24 = −27, which is not zero. So (−1, 28) is a non-stationary point of inflection.

Here x is the input and y = x³ + 3x² − 24x + 2. The curve is concave to the left of x = −1 and convex to the right. Every cubic has exactly one point of inflection, and it sits halfway between the two turning points when the cubic has them: halfway between x = −4 and x = 2 is x = −1.

Stationary and non-stationary points of inflection

A stationary point of inflection has a gradient of zero: the curve flattens out for an instant but keeps going the same way, like y = x³ at the origin. The gradient has the same sign on both sides. A non-stationary point of inflection has a gradient that is not zero: the curve is still rising or falling as its bend changes, as in Example 1.

Example 2: both kinds on one curve

Find and classify the points of inflection on y = x⁴ − 8x³ + 18x².

f″(x) = 0 at x = 1 and x = 3. It is a positive quadratic, so it is positive for x < 1, negative between 1 and 3, and positive for x > 3. It changes sign at both, so both are points of inflection.

• At x = 1: y = 1 − 8 + 18 = 11 and the gradient is 4 × 1 × (−2)² = 16. (1, 11) is a non-stationary point of inflection.
• At x = 3: y = 81 − 216 + 162 = 27 and the gradient is 4 × 3 × 0² = 0. (3, 27) is a stationary point of inflection.

Here x is the input and y = x⁴ − 8x³ + 18x². The curve rises from its minimum at the origin, changes bend at (1, 11) while still climbing, then flattens out at (3, 27) and keeps climbing. Notice that (x − 3)² in dy/dx never changes sign, so the gradient is positive on both sides of x = 3. That is the sign of a stationary point of inflection.

When f″(x) = 0 is not a point of inflection

f″(x) = 0 on its own is not enough. Take y = x⁴. Its second derivative is 12x², which is 0 at x = 0. But 12x² is positive on both sides of 0, so the curve is convex on both sides and there is no change of bend. The origin is a minimum, not a point of inflection. This is why step 3, checking the sign change, matters.

Check your understanding.

Classifying stationary points

Points of inflection come up when you classify stationary points, the points where dy/dx = 0. The second derivative test tells you what each one is:

When f″(x) = 0 at a stationary point, work out the gradient just to the left and just to the right. Positive then negative is a maximum; negative then positive is a minimum; the same sign on both sides is a stationary point of inflection. Practise classifying stationary points in this question from the Calculus course.

Convex and concave sections of a curve

To find where a curve is convex or concave, solve an inequality in the second derivative.

Example 3: where is the curve concave?

For y = x⁴ − 8x³ + 18x² from Example 2, f″(x) = 12(x − 1)(x − 3). The curve is concave where f″(x) < 0, which is 1 < x < 3, and convex where f″(x) > 0, which is x < 1 or x > 3. The points of inflection are the boundaries between these sections.

Example 4: a curve with two points of inflection

Find the points of inflection ony = e^{-x^2}. Use the chain rule, then the product rule:

e^{-x^2}is never zero, so f″(x) = 0 when 4x² − 2 = 0, giving x = ±1/√2 ≈ ±0.707. The factor 4x² − 2 changes sign at both, so both are points of inflection, aty = e^{-rac{1}{2}}≈ 0.607. The gradient is not zero at either, so they are non-stationary.

This bell shaped curve is the shape of the normal distribution, whose points of inflection are at μ − σ and μ + σ: a standard deviation from the mean on each side. You can check any derivative in the Derivative Calculator.

Common mistakes

• Stopping at f″(x) = 0. Always check that f″(x) changes sign. y = x⁴ is the classic trap.
• Thinking every point of inflection is stationary. Most are not. Only check f′(x) = 0 to decide which kind it is.
• Mixing up the tests. A sign change in f′(x) means a turning point; a sign change in f″(x) means a point of inflection.
• Substituting into the wrong function. The y coordinate comes from the original equation, not from f′ or f″.
• Swapping convex and concave. Convex means f″(x) > 0, like a bowl that holds water.

Practice questions

1. Find the coordinates of the point of inflection on y = x³ − 9x² + 5, and say whether it is stationary.
2. Show that the curve y = x³ + 6x² + 12x has a stationary point of inflection, and find its coordinates.
3. Find where the curve y = x⁴ − 6x² is concave.
4. Explain why y = x⁶ has no point of inflection at the origin.

Answers: 1) f″(x) = 6x − 18 = 0 gives x = 3, y = −49; f′(3) = −27, so (3, −49) is non-stationary. 2) f′(x) = 3(x + 2)², which is 0 at x = −2 and positive on both sides; f″(x) = 6x + 12 changes sign at x = −2. The point is (−2, −8). 3) f″(x) = 12x² − 12 < 0 gives −1 < x < 1. 4) f″(x) = 30x⁴ is 0 at x = 0 but positive on both sides, so the curve stays convex: the origin is a minimum.

Practise with a short timed paper

Stationary points, second derivatives and curve sketching come up in A-level Maths calculus. The daily A-level Maths: Calculus paper is a short timed paper of 8 questions, with a new one every day.

Read next

Maths · CalculusDerivative calculator with steps: every rule shownFree derivative calculator with steps: each rule named, second and higher derivatives, the slope and tangent at any point, and partial derivatives in x.