Question

If f(x) = cos x + a²x + b is an increasing function for all values of x, find a. 

Answer 1

Given f(x) = cosx + a²x + b

⇒ f'(x) = –sinx + a²

Since f(x) is increasing, so f'(x) ≥ 0

⇒ a² – sinx ≥ 0

⇒ a² ≥ sinx

⇒ a² ≥ 1

⇒ a² –1 ≥ 0

⇒ (a + 1)(a – 1) ≥ 0

⇒ a ≤ – 1, a ≥ 1

⇒ a ∈ (–∞, –1] ∪ [1, ∞)