Question

If f(x) =$\frac{\sin \rightleftharpoons (2\pi\lbrack\pi^{2}x\rbrack)}{5 + \lbrack x\rbrack^{2}}$ ([.] denotes the greatest integer function), then f(x) is

• A. Discontinuous at some x
• B. Continuous at all x, but the derivative $f^{'}(x)$ doesn’t exist for some x
• C. f′(x) exists for all x, but f′′(x) doesn’t exist some x
• D. f′′(x) exist for all x

Answer 1

Answer: (D)

f′′(x) exist for all x

Answer 2

Since $\left\lbrack \pi^{2}x \right\rbrack$ is an integer whatever be the value of x and so 2π$\left\lbrack \pi^{2}x \right\rbrack$ is an integral multiple of π.

Thus, sin (2π[π² x]) = 0 and 5 + [x]² ≠ 0 for all x.

Hence f(x) = 0 ∀ x ∈ R.

Thus, f(x) is a constant function and so it is continuous and differentiable any number of times for all x ∈ R