Minimization of two variables

Question

1. (General) How can I find $$\min\limits_{\phi\leq x\leq \alpha , \ \beta \leq y \leq \delta}\{a+bx,c+dy\}$$ given values of $a,b,c,d,\phi, \alpha, \beta, \delta \in \mathbb{R}?$
2. (Specific) How can I find $$\min\limits_{0 \leq x , \ y \ \leq 1}\{.4664048+(2147.0588450)x,0.0313096+(91.2500009)y\}?$$

Thanks.

Answer 1

Each of the two terms is linear, so will be minimized at an endpoint, so there are at most four items to check.

For the specific question, since $b,d$ are each positive, they will be minimized when $x=y=0$, so the minimum is $$\min\{.4664048,0.0313096\}=0.0313096$$