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.

Eliminating $b,$ we have $$a^2+(2-a-c)^2+c^2=12$$
$$\implies 2a^2-2a(2-c)+2c^2-4c-8=0\iff a^2-a(2-c)+c^2-2c-4=0$$ which is a Quadratic Equation in $a$
As $a$ is real, the discriminant must be $\ge0$ 
$$\implies (2-c)^2-4(c^2-2c-4)\ge0$$
$$\implies -3c^2+4c+20\ge0\iff3c^2-4c-20\le0$$
Now, we know if $(x-\alpha)(x-\beta)\le0$ with $\alpha\le\beta$
we can write $\alpha\le x\le\beta$
If $\alpha,\beta(>\alpha)$ are the roots of $3c^2-4c-20=0,$ we need $\displaystyle\beta-\alpha=+\sqrt{\left(\beta+\alpha\right)^2-4\beta\alpha}$
we have $\displaystyle\beta+\alpha=\frac43$ and $\displaystyle\beta\alpha=-\frac{20}3$
