Question

Use Rolle’s Theorem to prove that if $f$is continuous and differentiable everywhere and has three roots, then its derivative $f$ has at least two roots.

Short answer

We have proved using Rolle's Theorem that the derivative $f\text{'}$has at least two roots.

Step-by-step solution

Step 1. Given Information.

$f$is continuous and differentiable everywhere and has three roots.

Step 2. Using Rolle's Theorem.

Let the three roots of the function $f$be $r_1,r_2,$and $r_3$. Here, $f$is not continuous and differentiable everywhere. Rolle's Theorem guarantees that role="math" localid="1648530843246" $f\text{'}$will have at least one root on the interval $[r_1,r_2]$and at least one root on $[r_2,r_3]$.

Hence, the derivative of the function$f\text{'}$has at least two roots.