Optimization with a few Variables (AMC 12 question) In the 2013 AMC 12B, question 17 says:
Let $a$,$b$, and $c$ be real numbers such that
$a+b+c=2$, and $a^2+b^2+c^2=12$
What is the difference between the maximum and minimum possible values of $c$?
I was wondering if there is a quick and easy solution using multivariable calculus for this problem. (I've only taken single variable)
The only solution I've seen uses the Cauchy-Schwarz inequality.

Both extremes occur when $a=b.$
This is a geometric observation, the plane $a=b$ is a plane of symmetry for both the sphere and the (orthogonal) plane $a+b+c=2.$ Put another way, switching $(a,b)$ does not change anything, you still get a point in the intersection.
So, there is an argument that makes smaller demands on background. The intersection of a plane and sphere is a circle. For any point $(a,b,c),$ whenever $a \neq b,$ there are two points with the same $c$ value, $(a,b,c)$ and $(b,a,c).$ Such points are not along the circle diameter along which the maximum and minimum values of $c$ occur. So, the diameter has $a=b.$ Substitution then gives max $c = 10/3,$ min $c=-2.$ 
Oh, well. By carefully plugging in, one may confirm that this gives your circle as a function of an independent variable $t$
$$  a = \frac{2}{3} -   \frac{4}{\sqrt 3} \cos t -   \frac{4}{ 3} \sin t,    $$
$$  b = \frac{2}{3} +   \frac{4}{\sqrt 3} \cos t -   \frac{4}{ 3} \sin t,    $$
$$  c = \frac{2}{3} +    \frac{8}{ 3} \sin t,    $$
as $a+b+c=2$ and $a^2 + b^2 + c^2 = 12.$ As $\sin t$ varies between $1$ and $-1,$ the largest $c$ gets is $10/3,$ the smallest $-6/3 = -2.$ The extremes occur when $\cos t = 0,$ so $a=b.$
The information used to construct the parametrization above comes from the upper left three rows by three columns of
$$    
 \left(  \begin{array}{rrrrrrrrrr}
  1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1   \\
  1  &  1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1   \\
  1  &  0  &  2  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1   \\
  1  &  0  &  0  &  3  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1   \\
  1  &  0  &  0  &  0  &  4  &  -1  &  -1  &  -1  &  -1  &  -1   \\
  1  &  0  &  0  &  0  &  0  &  5  &  -1  &  -1  &  -1  &  -1   \\
  1  &  0  &  0  &  0  &  0  &  0  &  6  &  -1  &  -1  &  -1   \\
  1  &  0  &  0  &  0  &  0  &  0  &  0  &  7  &  -1  &  -1   \\
  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  8  &  -1   \\
  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  9   
\end{array}
  \right),
  $$
with columns that are orthogonal, but not orthonormal, eigenvectors for the matrix with all entries equal to $1.$
