Question:
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1]
Solution:
Given:
$f(x)=|x|$
If Lagrange's theorem is applicable for the given function, then $f(x)$ is continuous on $[-1,1]$ and differentiable on $(-1,1)$.
But it is known that $f(x)=|x|$ is not differentiable at $x=0 \in(-1,1)$.
Thus, our supposition is wrong.
Therefore, Lagrange's theorem is not applicable for the given function.