Question

Discuss the applicability of Rolle's theorem in the interval [-1,1] to the function f(x)=$\left|x\right|$.

Answer 1

We have $f\left(x\right)=\left|x\right|$

$\Rightarrow f(-1)=1$ and $f\left(1\right)=1$

$\Rightarrow f(-1)=f\left(1\right)$

Now the function $f\left(x\right)$ is continuous throughout the closed interval $(-1,1)$

but $f\left(x\right)$ is not differentiable at $x=0\in (-1,1)$

Hence, Rolle's Theorem is not satisfied. (due to the second condition)