Question

Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1]

Answer 1

Given:
$f\left(x\right)=\left|x\right|$

If Lagrange's theorem is applicable for the given function, then $f\left(x\right)$ is continuous on $\left[-1,1\right]$ and differentiable on $\left(-1,1\right)$.

But it is known that $f\left(x\right)=\left|x\right|$ is not differentiable at $x=0\in \left(-1,1\right)$.

Thus, our supposition is wrong.
Therefore, Lagrange's theorem is not applicable for the given function.