Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$ 
If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum  and minimum values of $a+b+c$.

How to solve this one?
(Here's the source of inspiration for the problem.)

Here's a geometric way of looking at it... the points where $x,y,z > 0$ where $x^2 + y^2 + z^2 = 27$ is the "upper right" 1/8 of the outside of the sphere centered at the origin, of radius $3\sqrt{3}$. Call this surface $S$. If you connect the corners $(3\sqrt{3},0,0)$, $(0,3\sqrt{3},0)$, and $(0,0,3\sqrt{3})$ of $S$, you get a triangle in the plane with equation $x + y + z = 3\sqrt{3}$. If you replace $3\sqrt{3}$ by any $r < 3\sqrt{3}$, then the plane $x + y + z = r$ will no longer intersect $S$. So the minimum is $3\sqrt{3}$.
Similarly, the plane $x + y + z = 9$ intersects $S$ at $(3,3,3)$ only, and replacing $9$ by any $r > 9$ will result in a plane   $x + y + z = r$ that doesn't intersect $S$ at all. So the maximum is $9$.
