Q. 67 - Calculus

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.

Step-by-Step Solution

Answer

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

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 $f\text{'}$ 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.