Question

Let f(x) be a function such that; f'(x) = log_1/3(log₃(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.

Options

• (1, 4)

• (4, ∞)

• (2, 3)

• (2, ∞)

Answer 1

Let f(x) be a function such that; f'(x) = log_1/3(log₃(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is (4, ∞).

Explanation:

f’(x) = log_1/3(log₃(sin x + a)) < 0; x∈R

⇒ `log_3(sinx + a) > (1/3)^circ`

⇒ log₃(sin x + a) > 1

⇒ (sin x + a) > 3¹

⇒ (sin x + a) > 3

Since, –1 ≤ sin x ≤ 1

⇒ –1 + a > 3 and 1 + a > 3

⇒ a > 4 and a > 2

⇒ a∈(4, ∞) and a∈(2, ∞)

Most appropriate answer is for all a∈(4, ∞)