# 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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