Question
Maximise subject to the constraints:
Solution — Step by Step
The constraints define a region in the xy-plane:
- (below or on the line )
- (right of the y-axis)
- (above the x-axis)
The feasible region is the triangle formed by the three boundary lines.
We find the vertices by solving pairs of boundary equations:
- Origin: — intersection of and
- A: — intersection of and : solve to get
- B: — intersection of and : solve to get
The three corner points are , , and .
By the Corner Point Theorem, the maximum of a linear objective function over a bounded feasible region occurs at one of the corner points.
| Corner Point | |
|---|---|
The largest value is at the point .
Maximum value of , achieved at , .
Why This Works
The Corner Point Theorem tells us that a linear function over a convex polygon can only attain its maximum (or minimum) at a vertex of the polygon. This happens because linear functions have no “curves” — they can’t have an interior maximum.
The coefficient of (which is 4) is larger than the coefficient of (which is 3), so the objective function grows faster in the -direction. This is why the maximum occurs at , where is as large as possible within the feasible region.
Alternative Method — Iso-Profit Line
Draw lines for increasing values of . These are parallel lines. As we push this line as far as possible while still touching the feasible region, the last corner it touches gives the maximum.
Moving the iso-profit line away from the origin, the last feasible corner touched is , confirming .
When the objective function has coefficients and for and respectively, and the only constraint is , you always want to maximize the variable with the higher coefficient. Here has coefficient 4 > 3, so put all resources into .
Common Mistake
Students often forget to check the origin as a corner point. While the origin gives (minimum here), neglecting it can lead to an incomplete table and wrong answers in problems where you need the minimum. Always list ALL corner points — origin included.