Sign of the derivative and direction of change

The fundamental relationship

Let f be a differentiable function on an interval I.

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

This theorem is the main tool for studying the behaviour of a function.

Method

1. Calculate f'(x).
2. Examine the sign of f'(x) (factorise if necessary; use a sign table).
3. Deduce the direction of variation of f, then draw up the table of variations.

Example: f(x) = x² - 4x + 3, defined on ℝ. f'(x) = 2x - 4 = 2(x - 2) f'(x) < 0 for x < 2, and f'(x) > 0 for x > 2. f'(2) = 0.

Table of variations

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

We calculate f(2) = 4 - 8 + 3 = -1: this is the minimum of f on ℝ.

Common pitfall

The sign of f' is determined over entire intervals, not just at a single isolated point. A point where f'(x) = 0 is not always a point of inflection: for example, for f(x) = x^3, f'(x) = 3x^2 ≥ 0 on R and is zero only at 0, but f remains strictly increasing on the entire set of R.