Question

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

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

Answer 1

The correct option is A $2a$

$f\left(x\right)=|x^2-a^2|$ for $a>0$



From the graph of $f\left(x\right)$ the longest interval in which Role's theorem is applicable is $[-a,a]$
because $f\left(x\right)$ is continuous in the closed interval $[-a,a]$ and $f\left(x\right)$ is differentiable in the in the open interval $(-a,a)$
So, length of longest interval is $a-(-a)=2a$