Solutions

i) $\sup = 1 = \max$. $\inf = 0$. The minimum does not exist.

ii) $\inf = \min = -1$. $\sup = \max = 1$.

iii) $\inf = \min = 0$. $\sup = \max = \frac{1}{n}$.

iv) $\sup = \sqrt{2}$. No maximum. $\inf = \min = 0$.

v) This set is $\mathbb{R}$ since for all $x$, $x^2 + x + 1 \ge 0$. No $\sup$, $\inf$, $\min$, or $\max$.

vi) This set is

$$\left\{x : \frac{-1-\sqrt{5}}{2} < x < \frac{-1+\sqrt{5}}{2}\right\}.$$

The $\inf$ and $\sup$ values are the boundaries. There is no maximum or minimum.

vii) This set is

$$\left\{x : \frac{-1-\sqrt{5}}{2} < x < 0\right\}.$$

The $\inf$ and $\sup$ values are at the boundaries. There is no maximum or minimum.

viii) $\inf = -1$. $\sup = \max = \frac{3}{2}$. No minimum.