Once you have the feasible region, the best value of the objective function is found by calculating it at every vertex and comparing. The vertex with the largest value gives the maximum, and the vertex with the smallest value gives the minimum.
This lesson follows drawing the feasible region in the linear programming chapter.
What do I do at each vertex?
Write the vertices in a table. Substitute each pair (x, y) into the objective function and record the value. Then read off the largest and smallest.
Worked example: a maximum and a minimum
A feasible region is given by x + y ≤ 10, x ≤ 6 and y ≥ 2, with x ≥ 0. The profit is P = 4x + 3y. Find the maximum and minimum of P.
First find the vertices:
- y = 2 and x = 0: (0, 2).
- y = 2 and x = 6: (6, 2).
- x = 6 and x + y = 10: (6, 4).
- x = 0 and x + y = 10: (0, 10).
Now test them:
| Vertex (x, y) | 4x + 3y | P |
|---|---|---|
| (0, 2) | 0 + 6 | 6 |
| (6, 2) | 24 + 6 | 30 |
| (6, 4) | 24 + 12 | 36 |
| (0, 10) | 0 + 30 | 30 |
The maximum is 36 at (6, 4) and the minimum is 6 at (0, 2).
When two vertices tie
Change the profit to P = 3x + 3y. The values become 6, 24, 30 and 30. The vertices (6, 4) and (0, 10) both give 30.
That means every point on the edge x + y = 10 between them gives 30. If the context needs whole numbers, (6, 4), (5, 5), (4, 6) and so on all give the maximum.
The mistake that costs marks
The first error is testing a point that is not a vertex, or an intersection that lies outside the region. The lines y = 2 and x + y = 10 meet at (8, 2), but x = 8 breaks x ≤ 6, so (8, 2) is not a vertex.
The second error is stopping at the vertex that looks furthest from the origin. For a minimum question that vertex may be the wrong one, so always test every vertex.
Check yourself
The region is given by x ≥ 1, y ≥ 2 and 2x + y ≤ 14. The cost is C = 5x + 2y. Find the minimum and maximum of C.
Answer
The vertices are:
- x = 1 and y = 2: (1, 2).
- y = 2 and 2x + y = 14: 2x = 12, so (6, 2).
- x = 1 and 2x + y = 14: y = 12, so (1, 12).
Values of C: (1, 2) gives 5 + 4 = 9. (6, 2) gives 30 + 4 = 34. (1, 12) gives 5 + 24 = 29.
The minimum is 9 at (1, 2) and the maximum is 34 at (6, 2).
What to study next
The last skill is writing the result in words. Continue with explaining a maximum or minimum in context. The word-problem structure worksheet helps keep the question’s labels straight.
If arithmetic at the vertices is where marks slip, see online one-to-one Additional Mathematics tuition.