A headhunter is trying to lure you away from your studies. He knows that you currently study 7 hours per day, and spend 3 hours outdoors per day, giving a total utility of $u(h, o) = u(7, 3)$. It costs the headhunter $c(h, o)$ to deliver $h$ hours of work and $o$ outdoor hours in the competing offer. If $u$ and $c$ are continuous on $X = \{(h, o) \in [0, 24]² : h + o \le 24\}$, is there a solution to the problem $\begin{aligned} &\min_{(h, o) \in X} c(h, o) \\ &\text{s.t. } u(h, o) \ge u(7, 3)? \end{aligned}$

The government chooses a tax policy $t \in CB(\mathbb{R}_+)$ to maximise social welfare $W : CB(\mathbb{R}_+) → \mathbb{R}$, where distances in the domain and co-domain are measured with $d_\infty$ and $d_2$. Suppose $W$ is continuous. Does the extreme value theorem imply that there is an optimal tax policy?

Suppose there are $10,000 m²$ of land and $1000L$ of water. These can be allocated to wine and to blueberry production. If $(l_v, w_v)$ is allocated to wine and $(l_b, w_b)$ is allocated to blueberry production, then the total output is $f(l_v, w_v)$ and $g(l_b, w_b)$, where $f$ and $g$ are continuous. Is the feasible set of possible wine and blueberry outputs compact?