# Thread: Prove the maximum number of critical points.

1. ## Prove the maximum number of critical points.

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

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

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

2. Originally Posted by cgiulz
Let $f(x)$ be a polynomial of degree $n.$

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

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

A critical point of a function is where its first derivative vanishes, so...

Tonio