Question

Let f : [0, ∞) → [0, ∞) and g : [0, ∞) → [0, ∞) be non-increasing and non-decreasing functions, h(x) = g(f(x)). If 'f ' and 'g' are differentiate for all points in their respective domains and h(0) = 0 then, h(x) will always be

• A. An increasing function
• B. A decreasing function
• C. Identically zero
• D. None of these

Answer 1

Answer: (C)

Identically zero

Answer 2

h(x) = g(f(x)) since g(x) ∈ [0, ∞)

⇒ h (x) > 0 ∀ x ∈ domain.

Also h'(x) = g'(f(x)). f'(x) ≤ 0 as g'(x) ≥ 0 and f '(x) ≤ 0

⇒ h(x) is non-increasing. Thus h(x) ≤ 0 ∀ x ∈ domain as h(0) = 0. Hence h(x) = 0 ∀ x ∈ domain.