Check the convexity of the following subsets of R$^n$.
I. $S_1 = \{(x_1, x_2) \in \mathbb{R}^2 : x_1 x_2 \ge 1, x_1 \ge 0, x_2 \ge 0\}$
II. $S_2 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : 2x_1 + 3x_2 + 4x_3 = 14\}$
III. $S_3 = \{(x_1, x_2) \in \mathbb{R}^2 : 4x_1^2 + 9x_2^2 \le 36\}$
IV. $S_4 = \{(x_1, x_2) \in \mathbb{R}^2 : x_1 x_2 \le 1, x_1 \ge 0, x_2 \ge 0\}$
V. $S_5 = \{(x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 \le 4\}$
