The number of points, at which the function $$f(x) = \max\{6x, 2 + 3x^2\} + |x - 1|\cos\left|x^2 - \frac{1}{4}\right|$$, $$x \in (-\pi, \pi)$$, is not differentiable, is _____.

$$g(x) = |x - 1| \cos|x^2 - \frac{1}{4}|$$

Since $$\cos|u| = \cos u$$, the cosine term is perfectly smooth everywhere.

The only sharp corner comes from $$|x - 1|$$ at $$x = 1$$. Since the cosine part doesn't become zero there ($$\cos(3/4) \neq 0$$), the function is non-differentiable at $$x = 1$$.

$$h(x) = \max\{6x, \, 2 + 3x^2\}$$

$$3x^2 - 6x + 2 = 0 \implies x = 1 \pm \frac{1}{\sqrt{3}}$$

At these two intersection points, the graph forms sharp corners as it switches from the line to the parabola. This creates 2 more points of non-differentiability.