Intervals from the sign of f′

Method: find f′, solve f′(x) = 0, mark those points on the number line, and test the sign of f′ in each piece. Positive pieces are where f increases, negative pieces are where it decreases.

Worked example: f(x) = x² − 4x + 6. f′ = 2x − 4 is zero at 2, negative on (−∞, 2) and positive on (2, ∞).

Worked example: f(x) = 4x³ − 6x² − 72x + 30. f′ = 12(x − 3)(x + 2), zero at −2 and 3. f increases on (−∞, −2) and (3, ∞) and decreases on (−2, 3).

Worked example: sin 3x on [0, π/2]. f′ = 3 cos 3x is zero at x = π/6; f increases on [0, π/6] and decreases on [π/6, π/2].

Worked example: sin x + cos x on [0, 2π]. f′ = cos x − sin x is zero at π/4 and 5π/4. f increases on [0, π/4) and (5π/4, 2π] and decreases on (π/4, 5π/4).

Worked example: f(x) = (3/10)x⁴ − (4/5)x³ − 3x² + (36/5)x + 11 has f′ = (6/5)(x − 1)(x + 2)(x − 3). It decreases on (−∞, −2) and (1, 3) and increases on (−2, 1) and (3, ∞).

A composite keeps the direction of its inside when the outside is increasing: tan⁻¹(sin x + cos x) rises on (0, π/4) because sin x + cos x does there.