Question

f(x) = $\int _ { 0 } ^ { x } \left| \log _ { 2 } \left( \log _ { 3 } \left( \log _ { 4 } ( \cos t + a ) \right) \right) \right| d t$ . If f(x) is increasing for all real values of x then

• A. a ∈ (-1, 1)
• B. a ∈ (1, 5)
• C. a ∈ (1, ∞)
• D. a ∈ (5, ∞)

Answer 1

Answer: (D)

a ∈ (5, ∞)

Answer 2

f (x) = |log₂ (log₃ (log₄ (cosx + a)))|. Clearly f(x) is increasing for all values of x if log₂ (log₃ (log₄ (cosx + a))) is defined for all values of x.

⇒ log₃ (log₄ (cosx + a)) > 0 ∀ x ∈ R

⇒ log₄ (cosx + a) > 1 ∀ x ∈ R

⇒ cosx + a > 4 ∀ x ∈ R

⇒ a > 5