The number of point where $|f(x)| + |x-2| - 1$ is non-differentiable in $x \in (0, 3\pi)$, where $f(x) = \prod_{k=1}^{n}\frac{\left(1+2\cos\left(\frac{2x}{3^k}\right)\right)}{3}$ is_____.
Step-by-Step Solution
Key Concept: Recognize the infinite product formula for $\sin x / x$ and identify points of non-differentiability by analyzing where the function or its derivative becomes discontinuous.
Start with $\sin 30° = 3\sin θ - 4\sin^3 θ = \sin θ(1 + 2\cos(2θ))$ where $θ = x/3^k$. Taking the limit as $n \to ∞$ and using the product formula, $\frac{\sin x}{x} = \prod_{k=1}^{∞} \frac{1 + 2\cos(2x/3^k)}{3} = f(x)$. The function $|f(x)| + |x-2| - 1$ is not differentiable at $\{π, 2π, 1, 2, 3\}$, giving 5 non-differentiable points.
Correct Answer: 2