Answer
$\left(\frac{5}{2},\sqrt {\frac{5}{2}}\right)$
Work Step by Step
The distance $d$ from the point $(3,0)$ to the point $(x,\sqrt x)$ on the line is given by
$d$ = $\sqrt {(x-3)^{2}+(\sqrt x-0)^{2}}$
and the square of the distance is
$S(x)$ = $d^{2}$ = ${(x-3)^{2}+(\sqrt x)^{2}}$ = $(x-3)^{2}+x$
$S'(x)$ = $2(x-3)(1)+1$ = $2x-5$
$S'(x)$ = $0$
$x$ = $\frac{5}{2}$
$S''(x)$ = $2$ $\gt$ $0$
so we know that $S$ has a minimum at $x$ = $\frac{5}{2}$
Thus the $y$-value is $\sqrt {\frac{5}{2}}$ and the point is $\left(\frac{5}{2},\sqrt {\frac{5}{2}}\right)$
