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.)

You can also use the method of Lagrange multipliers to maximize/minimize $f(a,b,c)=a+b+c$ given $g(a,b,c)=a^2+b^2+c^2=27$. We need the gradient of $f$ to be a multiple of the gradient of $g$, i.e., 
$$1=\lambda 2a,\quad 1=\lambda 2b,\quad 1=\lambda 2c,$$
where $\lambda$ is some real number. Hence:
$$\lambda = \frac{1}{2a}=\frac{1}{2b}=\frac{1}{2c}$$ 
and we must have $a=b=c$. This yields $g(a,a,a)=3a^2=27$, so that $a=b=c=3$ or $a=b=c=-3$. We clearly have a maximum at $a=b=c=3$. Since you assumed $a,b,c>0$, the minimum at $a=b=c=-3$ is not an allowed solution. Thus, the minimum must occur in the boundary of your domain (outside of your domain, so there is an infimum, but no minimum), i.e., when one at least on of $a,b,c$ is zero. However, say $c=0$, then we are trying to minimize $a+b$ given $a^2+b^2=27$. The min occurs when $a=b=-\sqrt{27/2}$ which is again outside our domain $a,b>0$, so the infimum occurs at the boundary when one of $a$ or $b$ is zero. And this leads to an infimum for $f(a,b,c)=a+b+c$ at $(\sqrt{27},0,0)$, $(0,\sqrt{27},0)$ and $(0,0,\sqrt{27})$, and the infimum value is $\sqrt{27}=3\sqrt{3}$. 
