Find the point on the cone closest to (1,4,0) Find the point on the cone $z^2=x^2+y^2$ nearest to the point $P(1,4,0)$.
This is a homework problem I've not made much headway on.

You want to minimize $\sqrt{(x-1)^2+(y-4)^2+z^2}$ where $x$, $y$, and $z$ satisfy $x^2+y^2=z^2$. Notice that if we minimize the distance squared, distance is also minimized, so it suffices to minimize $(x-1)^2+(y-4)^2+z^2$. We can use the constraint to substitute $x^2+y^2$ for $z^2$. On doing so, we get $$(x-1)^2+(y-4)^2+z^2,$$ which expands to $$2x^2+2y^2-2x-8y+17.$$ You can minimize this using usual gradient methods. You will get values for $x$ and $y$. Find $z$ by referring back to the constraint.
