Question

If the function ƒ(x) = $\lim_{m \rightarrow \infty}\left( \cos\frac{x}{m} \right)^{m}$ sin (x – 2) + a cos (x – 2), (where [.] denotes the greatest integer function) is continuous and differentiable in (4, 6), then –

• A. a ∈ [8, 64]
• B. a ∈ (0, 8]
• C. a ∈ [64, ∞)
• D. None of these

Answer 1

Answer: (C)

a ∈ [64, ∞)

Answer 2

We have, x ∈ (4, 6)

⇒ 2 < x – 2 < 4

⇒ $\frac { 8 } { a } < \frac { ( x - 2 ) ^ { 3 } } { a } < \frac { 64 } { a }$, a > 0

For ƒ(x) to be continuous and differentiable in (4, 6), $\left[ \frac { ( x - 2 ) ^ { 3 } } { a } \right]$ must attain a constant value for x ∈ (4, 6)

Clearly, this is possible only when a ≥ 64

In that case, we have

ƒ(x) = a cos (x – 2) which is continuous and differentiable

∴ a ∈ [64, ∞)

Hence, (3) is the correct answer.