Optimization Question Finding closest point Find the point on the graph of the function that is closest to the given point.
The function: $f(x)= \sqrt {x-8}$; the point: $(15,0)$
Given two points
$$
\cases{p_0=(15,0)\\
p=(x,\sqrt{x-8})}
$$
the squared distance is given by
$$
\|p-p_0\|^2= d^2
$$
or
$$
(x-15)^2+(\sqrt{x-8}-0)^2 = x^2-30x+15^2+x-8 = d^2
$$
but
$$
x^2-30x+15^2+x-8 = \left(x-\frac{29}{2}\right)^2+\frac{27}{4}=d^2
$$
so we can conclude that $d^2$ is minimum for $x=\frac{29}{2}$ and the minimal distance is $d=\frac{3\sqrt{3}}{2}$
Well, actually you can deal with this problem even without differentiating. You just expected to find the distance from any point on the graph of f(x) to (15; 0), minimize it and find out where the minimum is reached. (It can be done without using of derivative).