4. Let $f$ be the function given by $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + p$, where $p$ is an arbitrary constant. (a) Write an expression for $f ^ { \prime } ( x )$ and use it to find the relative maximum and minimum values of $f$ in terms of $p$. Show the analysis that leads to your conclusion. (b) For what values of the constant $p$ does $f$ have 3 distinct real roots? (c) Find the value of $p$ such that the average value of $f$ over the closed interval $[ - 1,2 ]$ is 1 . [Figure]
4. Let $f$ be the function given by $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + p$, where $p$ is an arbitrary constant.\\
(a) Write an expression for $f ^ { \prime } ( x )$ and use it to find the relative maximum and minimum values of $f$ in terms of $p$. Show the analysis that leads to your conclusion.\\
(b) For what values of the constant $p$ does $f$ have 3 distinct real roots?\\
(c) Find the value of $p$ such that the average value of $f$ over the closed interval $[ - 1,2 ]$ is 1 .\\
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