True or False: Between 2 sequential roots of $f'(x)$ there is at most one root for $f(x)$ I have a homework Question to answer which is:
True or False: Between 2 sequential roots of $f'(x)$ there is at most one root for $f(x)$
I think this is true since $f(x)$ would be monotone unless $f'(x)$ is equal to $0$ on an interval - but in that case what are 2 sequential values of the roots?
Can someone help me with this question? 
Thanks :)
 A: Suppose $f$ has two roots $x=a$ and $x=b$.  That is $f(a)=f(b)=0$. Assuming $f'$ exists on $[a,b]$, by the Mean Value Theorem (or Rolle's Theorem if you prefer), there is a $c\in(a,b)$ with $f'(c)=0$. 
This implies that if $c$ and $d$ are sequential roots of $f'$ and if $f'$ exists on $[c,d]$, then $f$ has at most one root between $c$ and $d$.

Added:

I am interpreting "between  two sequential roots of $f'\ $" to mean that there is a $c$ and a $d$ with $f'(c)=f'(d)=0$ and $f'(x)$ is non-zero for all $x$ between $c$ and $d$.
A: When $f'$ is not continuous then very strange things may occur. So let's assume that $f'$ is continuous, that $f'(a)=f'(b)=0$ and that $f'(x)\ne0$ for $a<x<b$. Then  $f'(x)>0$ for all $x\in\ ]a,b[\ $ or $f'(x)<0$ for all $x\in\ ]a,b[\ $. Assuming the former it follows that $f$ is strictly increasing on $[a,b]$, so there is at most one zero of $f$ in this interval.
A: Hint: http://en.wikipedia.org/wiki/Mean_value_theorem
A: It must be true, because the root of $f'(x)$ means it is maximum or minimum of $f(x)$, so the function in this interval is whether decreasing or rising, therefore there can't be more than one root of $f(x)$
Indeed, it can be a curving point. But even if it is a curving point, there is only 1 root in this interval.
