Tricky Points of Non-Differentiability

Calculus Level 3

f(x)=i=11+2cos(2x3i)3 f(x) = \prod_{i=1}^{\infty} \frac{1+ 2\cos\left( \frac{2x}{3^i} \right)}{3}

Let f(x)f(x) denote an function as described above. The number of points where xf(x)+x21 |xf(x)| + | |x-2|-1| is non-differentiable in x(0,3π) x \in (0,3\pi) is kk, find k2k^2.

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