Answer
(a) $f'$ has at least one root.
(b) $f''$ has at least one real root.
(c) Suppose $f$ is differentiable $n$ times on $\mathbb{R}$ and has $n+1$
roots. Then the $n$th derivative of $f$ has at least one real root.
Work Step by Step
(a) If $f$ is differentiable on $\mathbb{R}$ then $f$ is continuous on $\mathbb{R}$.
Suppose that $f$ has two roots $a_1$ and $a_2$, where $a_1 \lt a_2$.
Then $f(a_1) = f(a_2) = 0$.
According to Rolle's Theorem, there is a number $c_1$ in the interval $(a_1,a_2)$ such that $f'(c_1) = 0$
Therefore, $f'$ has at least one root.
(b) If $f$ is differentiable on $\mathbb{R}$ then $f$ is continuous on $\mathbb{R}$.
Suppose that $f$ has three roots $a_1$, $a_2$, and $a_3$, where $a_1 \lt a_2 \lt a_3$.
Then $f(a_1) = f(a_2) = f(a_3) = 0$.
According to Rolle's Theorem, there is a number $b_1$ in the interval $(a_1,a_2)$ such that $f'(b_1) = 0$, and there is a number $b_2$ in the interval $(a_2,a_3)$ such that $f'(b_2) = 0$.
Note that $f'(b_1) = f'(b_2) = 0$.
Since $f$ is twice differentiable on $\mathbb{R}$, then $f'$ is differentiable and continuous on $\mathbb{R}$.
According to Rolle's Theorem, there is a number $c$ in the interval $(b_1,b_2)$ such that $f''(c) = 0$.
Therefore, $f''$ has at least one real root.
(c) We can generalize parts (a) and (b):
Suppose $f$ is differentiable $n$ times on $\mathbb{R}$ and has $n+1$
roots. Then the $n$th derivative of $f$ has at least one real root.
