Roots Finding

Calculus Level 3

$\large f(x) = f(6-x) , f'(0) = f'(2) = f'(5) = 0.$

Consider all non-constant thrice differentiable functions $f$ defined for all reals $x$ that satisfies the above conditions. In the domain $0 \leq x \leq 6$ , what is the minimum number of roots to

$(f''(x))^2 + f'(x) \cdot f'''(x) = 0?$

Clarification: $f'(x), f''(x), f'''(x)$ denote the $1^\text{st}, 2^\text{nd} , 3^\text{rd}$ derivatives of the function $f(x)$ , respectively, with respect to $x$ .