Question

The length of the longest interval in which Rolle's theorem can be applied for the function $f\left(x\right)=|x^2-a^2|,(a>0)$ is

• A. $2a$
• B. $4a^2$
• C. $a\sqrt{2}$
• D. $a$

Answer 1

The correct option is A $2a$

According to Rolle's theorem: If $f$ be a continuous function on a closed interval $[A,B]$ and differentiable on the open interval $(A,B)$. If $f\left(A\right)=f\left(B\right)$, then there is at least one point $c$ in $(A,B)$ where $f^{\prime }\left(c\right)=0$.
Using the above information,
$f\left(x\right)=|x^2-a^2|$
$f\left(x\right)=|(x-a)(x+a)|$
Plotting $f\left(x\right)$, we can see that $f$ is continuous on $[-a,a]$, differentiable on $(-a,a)$ and also $f(-a)=f\left(a\right)$, and there also exists a point $c=0$ in $(-a,a)$ where $f^{\prime }\left(c\right)=0$.
So, the length of the longest interval is $2a$.