Question

Consider a function $f(x) = \prod_{r=1}^{\infty} \left( \frac{\sin \left( \frac{x}{3^{r-1}} \right)}{3 \sin \left( \frac{x}{3^r} \right)} \right)$. Then, the number of points where $|x f(x)|$ is non-differentiable on interval $(0, 3\pi)$ is _____.

Answer 1

2

Answer 2

The infinite product simplifies to $f(x) = \frac{\sin(x)}{x}$ for $x \neq 0$. Thus, $|x f(x)| = |x \cdot \frac{\sin(x)}{x}| = |\sin(x)|$ for $x \in (0, 3\pi)$. The function $|\sin(x)|$ is non-differentiable when $\sin(x) = 0$. In the interval $(0, 3\pi)$, $\sin(x) = 0$ at $x = \pi$ and $x = 2\pi$. There are 2 such points.