Minimum number of roots 1

Calculus Level 5

Let $$f(x)$$ be a thrice differentiable function satisfying:

$$|f(x) - f(4-x)| + |f(4-x)-f(4+x)| = 0, \forall x \in R$$

If $$f'(1)=0$$, then find the minimum number of roots of $$f'(x)\cdot f'''(x)+(f''(x))^2 =0$$, on $$x \in [0,6]$$.

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