It is rather easy to find extreme points in the $2$ variable case. How to find them for higher dimensions, say, in $3$ variables? For instance,
$$\begin{array}{ll} \text{minimize} & -3x_1-2x_2-x_3\\ \text{subject to} & 2x_1+x_2-x_3\le2\\ & x_1,x_2,x_3\ge0\end{array}$$
In your example, you can have arbitrary small values for your objective function. Just take $x_1=0,x_2=c,x_3=c$, then the constraint is satisfied for all $c\geq 0$, and you can make $c$ arbitrarily large, so your objective function $-3x_1-2x_2-x_3$ becomes arbitrarily small. In general you can use the simplex algorithm.