Increasing and decreasing functions

• A function f is increasing (ascending) on an interval when f′(x) > 0 at every point of it, and decreasing (descending) when f′(x) < 0.
• Procedure: compute f′(x), solve f′(x) = 0 for the critical numbers, then test the sign of f′ in each interval bounded by them (substitute a convenient value, or use the factor-sign chart).
• f is never increasing at a point where f′ does not exist (e.g. a cusp) — always state the domain first.

Worked example

f(x) = x³ − 6x² + 9x + 2 ⟹ f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3). f′ = 0 at x = 1, 3.

• x < 1: f′ > 0 (increasing) · 1 < x < 3: f′ < 0 (decreasing) · x > 3: f′ > 0 (increasing).
• So f increases on (−∞, 1) ∪ (3, ∞) and decreases on (1, 3).

The strictly-increasing condition

If f(x) = x³ + 3x² + kx is increasing on ℝ, then f′(x) = 3x² + 6x + k ≥ 0 for all x, so its discriminant ≤ 0: 36 − 12k ≤ 0 ⟹ k ≥ 3.