Tricky Points of Non-Differentiability

Calculus Level 4

\[ f(x) = \displaystyle \prod_{k=1}^{\infty} \left( \dfrac{1+ 2\cos\left( \dfrac{2x}{3^k} \right)}{3}\right) \]

Let \(f(x)\) denote an function as described above. The number of points where \( |xf(x)| + | |x-2|-1| \) is non-differentiable in \( x \in (0,3\pi) \) is \(k\), find \(k^2\).

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