Answer
$c=\dfrac{3}{2}$
Work Step by Step
Since, the given function $f(x)=(\sqrt {x-1})$ is continuous on $[1,3]$ and differentiable on $(1,3)$ and $f'(x)=\dfrac{1}{2\sqrt{x-1}}$ .
The Mean value Theorem states that there is a point $c$ and $c \in (1,3)$.
This means that $f'(c)=\dfrac{f(3)-f(1)}{(3-1)}=\dfrac{\sqrt {3-1}-\sqrt {1-1}}{(3-1)}=\dfrac{\sqrt 2}{2}$
This implies that $f'(c)=\dfrac{1}{2\sqrt{c-1}}=\dfrac{\sqrt 2}{2}$
Thus, $c=\dfrac{3}{2}$
