Using the derivative: sign and direction of change

The relationship between the sign of f' and the direction of change of f

This is one of the most useful results in the syllabus: the sign of the derivative provides direct information about the direction of change of f.

• If f'(x) ≥ 0 on an interval I (and f’ is zero only at isolated points), then f is increasing on I.
• If f'(x) ≤ 0 on an interval I, then f is decreasing on I.
• If f'(x) = 0 at a point a and f' changes sign around a, then f has a local extremum (maximum or minimum) at a, and the tangent at this point is horizontal.

Complete example

Let f(x) = x² - 4x + 1, with derivative f'(x) = 2x - 4.

We solve f'(x) = 0: 2x - 4 = 0, so x = 2.

x	-∞	2	+∞
sign of f'(x)	negative	0	positive
variation of f	decreasing	minimum	increasing

f therefore has a minimum at x = 2, which is f(2) = 4 - 8 + 1 = -3.

Key points to remember for practical application

General method for analysing the behaviour of a function:

1. Calculate f'(x).
2. Examine the sign of f'(x) (factorise, sign table, etc.).
3. Deduce the table of variations for f.

Common pitfall

f'(a) = 0 does not always mean that there is an extremum: if f' does not change sign around a (as with f(x) = x³ at x = 0), this is simply an inflection point with a horizontal tangent, not a maximum or minimum.