Roots Finding

Calculus Level 3

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

Consider all non-constant thrice differentiable functions ff defined for all reals xx that satisfies the above conditions. In the domain 0x6 0 \leq x \leq 6 , what is the minimum number of roots to

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

Clarification: f(x),f(x),f(x) f'(x), f''(x), f'''(x) denote the 1st,2nd,3rd1^\text{st}, 2^\text{nd} , 3^\text{rd} derivatives of the function f(x)f(x) , respectively, with respect to xx.

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