#### Answer

$(\frac{5}{2},\sqrt{\frac{5}{2}})$ is the point on the curve $y = \sqrt{x}$ that is closest to the point $(3,0)$

#### Work Step by Step

All the points on the curve $y = \sqrt{x}$ have the form $(x, \sqrt{x})$
Note that $x \geq 0$ and $y \geq 0$
We can write an expression for the distance from points on the curve to the point $(3,0)$:
$d = \sqrt{(x-3)^2+(y-0)^2}$
$d = \sqrt{(x-3)^2+(\sqrt{x})^2}$
$d = \sqrt{x^2-6x+9+x}$
$d = \sqrt{x^2-5x+9}$
We can find the point where $d'(x) = 0$:
$d(x) = \sqrt{x^2-5x+9}$
$d'(x) = \frac{2x-5}{2\sqrt{x^2-5x+9}} = 0$
$2x-5 = 0$
$x = \frac{5}{2}$
When $0 \leq x \lt \frac{5}{2}$, then $d'(x) \lt 0$
When $x \gt \frac{5}{2}$, then $d'(x) \gt 0$
Thus, $x=\frac{5}{2}$ is the point where $d(x)$ is a minimum.
We can find $y$:
$y = \sqrt{x} = \sqrt{\frac{5}{2}}$
$(\frac{5}{2},\sqrt{\frac{5}{2}})$ is the point on the curve $y = \sqrt{x}$ that is closest to the point $(3,0)$