Question

Examine the applicability of Mean Value Theorem for the following function.

$f\left(x\right)=[x$] for $x$ $ϵ$ $[2,2]$

Answer 1

Mean Value Theorem holds for a function $f:[a,b]\to R$, if following two conditions holds

(i) $f$ is continuous on $[a,b]$

(ii) $f$ is differentiable on $(a,b)$

Then, there exists some $c\in (a,b)$ such that

$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Mean Value Theorem is not applicable to those functions that do not satisfy any of the two conditions of the hypothesis.

Given function $f\left(x\right)=\left[x\right]$ for $x\in [-2,2]$
Since, the greatest integer function is not continuous at integral points.
So, $f\left(x\right)$ is not continuous at $x=\{-2,-1,0,1,2\}$

Hence, Mean Value theorem is not applicable to given function.