Direction of change and derivative

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

The central result of this chapter: the sign of the derivative provides direct information about the direction of variation of f over an interval.

• If f'(x) > 0 on an interval I, then f is strictly increasing on I.
• If f'(x) < 0 on an interval I, then f is strictly decreasing on I.
• If f'(x) = 0 on an interval I, then f is constant on I.

Method for analysing variations

1. Calculate f'(x).
2. Examine the sign of f'(x) (sign table, factorisation, etc.).
3. Deduce the table of variations for f.
4. Identify the local extrema: these occur where f' changes sign (and becomes zero).

Practical example

Let f(x) = x² - 4x + 3, defined on R. We calculate f'(x) = 2x - 4.

f'(x) = 0 <=> 2x - 4 = 0 <=> x = 2

f'(x) > 0 <=> 2x - 4 > 0 <=> x > 2, and therefore f'(x) < 0 for x < 2.

x	-∞	2	+∞
f'(x)	negative	0	positive
f(x)	decreasing	minimum: f(2) = -1	increasing

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

Common pitfall

A single change in the sign of f' at a point is sufficient for a local extremum, but if f' becomes zero without changing sign (for example, f(x) = x³ at x = 0, with f'(x) = 3x² ≥ 0 everywhere), there is NO extremum: the function remains increasing despite f'(0) = 0. Always check for a change in sign, not just whether the derivative is zero.