Sign of the derivative and direction of change

The fundamental relationship

The derivative f'(x) of a function f provides direct information about the direction of variation of f over an interval I:

• if f'(x) ≥ 0 on I (zero at at most isolated points), then f is increasing on I;
• if f'(x) ≤ 0 on I, then f is decreasing on I;
• if f'(x) = 0 throughout I, then f is constant on I.

Method

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

Example

Let f(x) = x² - 4x + 3. We have f'(x) = 2x - 4, which is zero at x = 2.

• For x < 2: f'(x) < 0, so f is decreasing.
• For x > 2: f'(x) > 0, so f is increasing.

Table of variations

x	-∞		2		+∞
f'(x)		-	0	+
f(x)	+∞ (decreasing)		-1		+∞ (increasing)

Common pitfall

A point where f'(x) = 0 is not necessarily a change in the direction of variation: the sign of f' must actually change at that point. For example, for f(x) = x³, f'(x) = 3x² ≥ 0 everywhere, so f is increasing on the entire real line, even though f'(0) = 0.