Second derivative, stationary points and optimisation

Maths: Analysis and Approaches · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The second derivative f″(x) is the derivative of f′(x). It tells you how the gradient is changing. Where f″(x) > 0 the gradient is increasing and the curve is concave up, like a cup. Where f″(x) < 0 the curve is concave down.

A stationary point is where f′(x) = 0. It is a local maximum if the gradient changes from positive to negative, and a local minimum if it changes from negative to positive. A point of inflexion is where the concavity changes, so f″(x) = 0 and f″ changes sign.

In optimisation, write the quantity as a function of one variable, set its derivative to zero, and check that the answer is a maximum or minimum.

Rules to remember

Common mistake

Saying that f″(x) = 0 is enough for a point of inflexion. For y = x4, y″ = 0 at x = 0 but y″ does not change sign, so (0, 0) is a minimum.

Worked example

Find and classify the stationary points of y = 2x3 − 3x2 − 12x + 4.

  1. dy/dx = 6x2 − 6x − 12 = 6(x − 2)(x + 1) = 0, so x = 2 or x = −1
  2. At x = 2: y = 16 − 12 − 24 + 4 = −16. At x = −1: y = −2 − 3 + 12 + 4 = 11
  3. d2y/dx2 = 12x − 6. At x = 2 it is 18 > 0, so a minimum
  4. At x = −1 it is −18 < 0, so a maximum

Answer: Local maximum at (−1, 11) and local minimum at (2, −16)

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