Answer
The theorem can not be applied because
$f$ is not differentiable on $[2, 6].$
Work Step by Step
The Mean Value Theorem
If $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ ,
then there exists a number $c$ in $(a, b)$ such that
$f^{\prime}(c)=\displaystyle \frac{f(b)-f(a)}{b-a}.$
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$f(x)=|5-x|=|x-5|$
this function's graph is the graph of $|x|$ translated 5 units to the right.
$|x|$ is not differentiable at $x=0$, so $f(x)$ is not differentiable at $x=5\in(2, 6).$
(Using the definition of the derivative at x=5,
the left limit of $\displaystyle \frac{f(x)-f(5)}{x-5} $is $-1$, while the right limit is $+1$,
The limit does not exist, so $f^{\prime}(5)$ is not defined)
The premises of the theorem are not satisfied, sot it can not be applied.
The theorem can not be applied because
$f$ is not differentiable on $[2, 6].$
