Let $\displaystyle f(x)$ be a polynomial of degree $\displaystyle n.$

$\displaystyle (a)$ What is the maximum number of critical points $\displaystyle f(x)$ can have? For each $\displaystyle n,$ give an example, with proof, of a polynomial having the maximum number for that $\displaystyle n.$

$\displaystyle (b)$ Answer the same question with "maximum" replaced with "minimum."