Question

If $f(x) = \sin(\sqrt{[\lambda]}x)$ is a periodic function with period $\frac{\pi}{2}$, where $[\lambda]$ denotes the greatest integer less than or equal to $\lambda$, then:

• A. $\lambda \in [4, 5)$
• B. $\lambda \in [4, 5]$
• C. $\lambda \in [16, 17)$
• D. $\lambda \in [16, 17]$

Answer 1

Answer: (C)

$\lambda \in [16, 17)$

Answer 2

The period of the function $f(x) = \sin(\sqrt{[\lambda]}x)$ is given by $T = \frac{2\pi}{\sqrt{[\lambda]}}$. We are given that the period is $\frac{\pi}{2}$. Therefore, we have the equation:

$\frac{2\pi}{\sqrt{[\lambda]}} = \frac{\pi}{2}$

Solving for $[\lambda]$:

$\sqrt{[\lambda]} = \frac{2\pi}{\frac{\pi}{2}} = 4$

$[\lambda] = 4^2 = 16$

Since $[\lambda] = 16$, we have $16 \le \lambda < 17$. Thus, $\lambda \in [16, 17)$.