Monotonicity

Monotonicity

A function f defined on an interval [a, b], said to be

(a) Non-decreasing function

if x² > x1 ⇒ f(x²) > f(x1), ∀ x1 x² ∈ [a, b] or f ′(x) > 0

(b) Increasing

if x² > x1 ⇒ f(x2) > f(x1), ∀ x1, x² ∈ [a, b] ⇒ f ′(x) > 0

(c) Non-Increasing

x² ≥ x1 ⇒ f(x²) ≤ f(x1), ∀ x1, x² ∈ [a, b]

⇒ f ′(x) ≤ 0

(d) Decreasing

x² > x1 ⇒ f(x²) < f(x1) ∀ x1, x² ∈ [a, b]    ⇒ f ′(x) < 0