0
$\begingroup$

I recently learned Rolle's Theorem and the Mean Value Theorem, and I was wondering if someone can please help me with the following:

Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Also suppose $f(x) = 0$ for $4$ values of $x$ in $(a, b)$. How can I show that there are at least two values of $x$ in $(a, b)$ for which $f'(x) = 0$?

The idea of $f$ being continuous and $[a, b]$ and differentiable on $(a, b)$ hints to using the Mean Value Theorem or Rolle's Theorem, but I'm not sure how to approach this problem. Any help is appreciated

$\endgroup$

1 Answer 1

0
$\begingroup$

Let $x_1,x_2,x_3,x_4 \in (a,b)$ with $f(x_i)=0$ and $x_1<x_2<x_3<x_4.$

By Rolle: there is $y_1 \in (x_1,x_2)$ such that $f'(y_1)=0.$

Can you proceed ?

$\endgroup$

You must log in to answer this question.