Question

Find the value of $c$ of Rolle's theorem for $f\left(x\right)=\left|x\right|$ in $[-1,1]$.

• A. $0$
• B. $1$
• C. $-1$
• D. does not exist

Answer 1

The correct option is D does not exist

Rolle's theorem states that if $f\left(x\right)$ be continuous on $[a,b]$, differentiable on $(a,b)$ and $f\left(a\right)=f\left(b\right)$ then there exists some $c$ between $a$ and $b$ such that $f^{{}^{\prime }}\left(c\right)=0$

Given $f\left(x\right)=|x|$ and $[a,b]=[-1,1]$

Rolle's theorem is not applicable for $f\left(x\right)$. Since, $f\left(x\right)$ is not differentiable at $x=0$ i.e., the graph of $|x|$ has a corner point at origin.

Therefore, $c$ value does not exist