Suppose that $A\left(5,6\right)$ and $B\left(3,4\right)$ be two points and P  be a point on  x- axis such that $\left|\mathrm{PA}+\mathrm{PB}\right|$ is minimum then the point P is

Suppose that  the images of $A\left(5,6\right)$ and $B\left(3,4\right)$ in the X-  axis are  C and D respectively. 
The point P  is the point of intersection of the lines  AD and  x- axis, it is coincide with the point of intersection of the lines  BC and x-  axis
The image of $B\left(3,4\right)$ in the x-  axis is  $\mathrm{D}\left(3,-4\right)$
The equation of the line joining $D\left(3,-4\right)$ an $A\left(5,6\right)$ is  $y-6=\frac{6+4}{5-3}\left(x-5\right)$

This can be simplified as 

$\begin{array}{c}y-6=\frac{6+4}{5-3}\left(x-5\right)\\ y-6=5\left(x-5\right)\\ y-6=5x-25\\ 5x-y-19=0\end{array}$

Plug in  x=0 to get the point of intersection of the above line and    axis. 
It implies that

$\begin{array}{c}5x=19\\ x=\frac{19}{5}\end{array}$

Therefore, $P=\boxed{\left(\frac{19}{5},0\right)}$