A region is given by x + y ≥ 4, x ≥ 0 and y ≥ 0…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A region is given by , and . Its corners are and . A learner wants the minimum of . Work out at the corners. Then work out at the feasible points and . Say whether a minimum exists, and explain in one sentence.
Need a reminder? The key ideaShow
On an unbounded region, let be the smallest corner value for a minimum. If the open half-plane has no point in common with the region, then is the minimum. If it has a common point, there is no minimum. Use and the largest corner value for a maximum.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The corners. At , . At , . So .
Step 2. Further out. The points and obey , so they are feasible. At , . At , . Both are below .
Step 3. Conclude. The half-plane has points in common with the region. Moving up the axis lowers without limit, so there is no minimum.
Answer: , and at the three points. No minimum exists, because the negative number in front of lets fall as grows.
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