Q. 65 - Calculus

Question

Prove Rolle’s Theorem: If $f$f is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f\left(a\right)=f\left(b\right)=0$, then there is some value $c\in (a,b)$ with $f\left(c\right)=0$.

Step-by-Step Solution

Answer

We have proved the statement is true for Rolle's Theorem.

Step 1. Given Information.

The function $f$ is  given.

Step 2. Rolle's Theorem.

If $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$ with $f\left(a\right)=f\left(b\right)=0$. We know that $f$ attains both a maximum and a minimum value on $[a,b]$. If one of these extreme values occur at $x=c$ in the interior $(a,b)$ of the interval, then $x=c$ is a local extremum of $f$. Using the theorem of "Local Extrema are Critical Points" this means that $x=c$ is a critical point of $f$. Hence, it follows that $f\text{'}\left(c\right)=0$ as $f$ is assumed to be differentiable at $x=c$. Hence, Rolle's Theorem is verified.