So many values are already given

Calculus Level 3

Let \(f(x)\) be a non-constant thrice differentiable function defined on real numbers such that \(f(x)=f(6-x)\) and \(f'(0)=0=f'(2)=f'(5)\). Find the minimum number of values of \(p \in [0,6]\) which satisfy the equation \[(f''(p))^2+f'(p)f'''(p)=0\] Details and Assumptions:

  • \(f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}\)

  • \(f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}\)

  • \(f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}\)

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