Let $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 2$. Compute the maximum value of $a + 2b + 3c$.
I have been solving similar problems using Cauchy-Schwarz. However, I do not know how to start this problem. Do I write $a^2=2-b^2-c^2$ or $a^2+b^2=2-c^2$ and work from there, or do I start using something else.
$a^2+b^2+c^2=2$ is a sphere of radius $\sqrt 2$ around the origin. The values of $a+2b+3c$ are constant on planes perpendicular to $(1,2,3)$. Find the point in the first octant where $a+2b+3c=k$ is tangent to the sphere and you are there.