Cubic polynomials give third degree burns to my brain

Calculus Level 4

Let \(\displaystyle f(x)\) be a polynomial of degree \(\displaystyle 3\), satisfying \(\displaystyle f(3) = 5\) and \(\displaystyle f(-1) = 9\).

\(\displaystyle f(x)\) has a minimum at \(\displaystyle x=0\) and \(\displaystyle f'(x)\) has a maximum at \(\displaystyle x=1\).

Find the distance between the local maximum and local minimum of \(\displaystyle f(x)\).

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