Roots Finding

Calculus Level 4

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

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 \(x\).

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