Optimization Question Finding closest point

Question

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: $(15,0)$

Answer 1

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 $d=\frac{3\sqrt{3}}{2}$

Answer 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).