Local extrema of a function

Definitions

We say that f has a local maximum at x₀ if there exists an interval around x₀ on which f(x) ≤ f(x₀). Similarly, f has a local minimum at x₀ if f(x) ≥ f(x₀) in the neighbourhood of x₀.

Necessary condition

If f is differentiable at x₀ and has a local extremum at x₀, then f'(x₀) = 0. Note that the converse is false: f'(x₀) = 0 does not guarantee an extremum (see the pitfall below).

Practical method

To find local extrema:

1. Solve f'(x) = 0 to obtain the candidate points.
2. Examine the sign of f' on either side of each candidate point.
3. If the sign changes, it is a local extremum (maximum if the sign changes from + to –, minimum if it changes from – to +). Otherwise, it is an inflection point.

Example

Let f(x) = x³ - 3x. We have f'(x) = 3x² - 3 = 3(x-1)(x+1), which is zero at x = -1 and x = 1.

• For x < -1: f' > 0 (increasing).
• Between -1 and 1: f' < 0 (decreasing).
• For x > 1: f' > 0 (increasing).

The sign changes at x = -1 (local maximum, f(-1) = 2) and at x = 1 (local minimum, f(1) = -2).

Pitfall

For f(x) = x³, f'(x) = 3x² is zero at x = 0, but the sign of f' does not change (positive before and after). There is therefore no extremum at x = 0, only an inflection point: always check for a change in sign, not just whether f' is zero.