Monotonic function

Strictly increasing function: A function f (x) is said to be a strictly increases function on (a, b) if x₁ < x₂ ⇒ f (x₁) < f (x₂) ∀ x₁, x₂ ϵ (a, b)

Ex: aˣ is strictly increasing function on R for a > 1

F(x) = eˣ is increasing on R

Strictly decreasing function: A function f (x) is said to be a strictly decreasing function on (a, b) if x₁ < x₂ ⇒ f (x₁) > f (x₂) ∀ x₁, x₂ ϵ (a, b)

F (x) = aˣ (0 < a < 1) is strictly decreasing function

Monotonic function: A function f(x) is said to be monotonic on an interval (a, b) if it is either increasing or decreasing (a, b)

Definition 1: A function f(x) is said to be increasing (decreasing) at a point x₀, if there is an interval (x₀ - h, x₀ + h) containing  x₀ such that f(x) is increasing (decreasing) on (x₀ - h, x₀ + h)

Definition 2:  A function f(x) is said to be increasing (decreasing) on [a, b] if it is increasing (decreasing) on (a, b) and it is also increasing (decreasing) on (a, b) and it is also increasing (decreasing) at x = a and x = b

Necessary and sufficient condition Monotonicity of functions:

Necessary condition: Let f(x) be a differentiable function defined on (a, b) then f’(x) > 0 or f’(x) < 0 according as f(x) is increasing or decreasing on (a, b)

If f(x) is an increasing function on (a, b) then tangent at every point on curve y = f(x) makes an acute angle in the positive direction of x – axis

tan θ > 0 ⇒ dy/dx > 0 Or f’ (x) > 0 ∀ x ϵ (a, b)

Similarly for decreasing function on (a, b) then the tangent at every point on the curve y = f(x) makes obtuse angle θ with direction of x - axis

tan θ < 0 ⇒ dy/dx > 0 Or f’ (x) < 0 ∀ x ϵ (a, b)

Sufficient condition: Let f be a differentiable real function defined on open interval (a, b)

a) If f’(x) > 0 ∀ x ϵ (a, b), then f is Increasing on (a, b)

b) If f’(x) < 0 ∀ x ϵ (a, b), then f is decreasing on (a, b)

Properties of Monotonic functions:

i) If f(x) is strictly increasing function on an interval [a, b] then f⁻¹ exists and also a strictly increasing function.

ii) If f(x) is strictly increasing function on an interval [a, b] such that it is continuous then f⁻¹ is continuous on [f (a), f (b)].

iii) If f(x) is continuous on [a, b] such that f’< (c) ≥ 0 (f’ (c) < 0) ∀ x ϵ (a, b) then f(x) is monotonically (strictly) increasing function on [a, b].

iv) If f(x) and g(x) are monotonically (or strictly) increasing (or decreasing) function on [a, b], then gof(x) is a monotonically (or strictly) increasing function on [a, b].

v) If one of the two function f(x) and g(x) is strictly (or monotonically) increasing and other a strictly (monotonically) decreasing, the gof (x) is strictly (monotonically) decreasing on [a, b].

If f(x) is monotonically increasing function ∀ x ϵ R such that f’’ (x) > 0 and [f⁻¹ (x₁) + f⁻¹ (x₂) + f⁻¹ (x₃)]/3 < f⁻¹ (x₁ + x₂ + x₃)/3.