Question

find the point on the graph of f(x)=sqrt(x-8) that is closest to the point (12,0)

Answer 1

Let \((a,b)=(a,\sqrt{a-8})\) be the point on the curve.

The distance between \((a,b)\) and \((12,0)\) will be
\[\begin{align*}d&=\sqrt{(a-12)^2+(b-0)^2}\\
&=\sqrt{(a-12)^2+(\sqrt{a-8})^2}\\
&=\sqrt{a^2-24a+144+(a-8)}\\
&=\sqrt{a^2-23a+136}
\end{align*}\]
Use the first derivative test to figure out the \(a\) that minimizes \(d\).

One thing to note is that, given \(f(x)=\sqrt{g(x)}\), \(f\) is optimized for the same \(x\) as \(g\) is, which means you can use the derivative test on
\[d^*=a^2-23a+136\]
to solve the problem.