Question 1 Foundation
What is the feasible region?
Answer. The set of points satisfying every constraint simultaneously.
It is the overlap of all the half-planes, and the optimum always lies at one of its vertices.
Linear Programming
Original questions written at exam standard. Work each one before you open the solution.
Difficulty
Question 2 Foundation
Where does the optimum of a linear programming problem occur?
Answer. At a vertex of the feasible region.
Sliding the objective line as far as possible leaves it touching a corner — or, if it is parallel to an edge, that whole edge.
Question 3 Standard
Translate "at least three times as many as " into an inequality.
Answer.
Test it: if then must be at least 6, which gives. Writing reverses the meaning.
Question 4 Standard
Find where and intersect.
Answer.
Subtracting the first from the second gives , and then .
Question 5 Stretch
Evaluate at , and , and state the maximum.
Answer. , , — the maximum is 36 at .
Every vertex must be tested; the largest value may be at an interior corner rather than an axis intercept.
Question 6 Stretch
Why is rounding a fractional optimum to the nearest integer unreliable?
Answer. The rounded point may lie outside the feasible region, or not be the best integer point.
Test the nearby integer points that satisfy every constraint, and compare their objective values.