$\left(i\right)\, f \left(x\right)=\sqrt{x}$ is continuous in $\left[4, 9\right]$
$\left(ii\right)\, f '\left(x\right)=\frac{1}{2\sqrt{x}}$
Thus $f\left(x\right)$ is differentiable in $\left(4, 9\right)$
$\left(iii\right)\, f \left(4\right)\ne f \left(9\right)$. All the three conditions of $LMV$ theorem satisfied then there exist at least one $c\,\in\left(4, 9\right)$ such that.
$f '\left(c\right)=\frac{f \left(b\right)-f \left(a\right)}{b-a} \Rightarrow \frac{1}{2\sqrt{c}}=\frac{1}{5} \Rightarrow c=\frac{25}{4}$