Define what it means to say that a set $S \subseteq \mathbb{R}^{n}$ is convex. What is meant by an extreme point of a convex set $S$ ?

Consider the set $S \subseteq \mathbb{R}^{2}$ given by

$$S=\left\{\left(x_{1}, x_{2}\right): x_{1}+4 x_{2} \leqslant 30,3 x_{1}+7 x_{2} \leqslant 60, x_{1} \geqslant 0, x_{2} \geqslant 0\right\}$$

Show that $S$ is convex, and give the coordinates of all extreme points of $S$.

For all possible choices of $c_{1}>0$ and $c_{2}>0$, find the maximum value of $c_{1} x_{1}+c_{2} x_{2}$ subject to $\left(x_{1}, x_{2}\right) \in S$.