The corner points of the feasible region determined by system of linear constraints are (60, 0), (120, 0), (40, 20) and (60, 30). Let \(z = ax + by\), \(a, b \gt 0\) be the objective function. Find condition on a and b so that the maximum of \(z\) occurs at (120, 0) and (60, 30).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2a=b\)
If the maximum occurs at two corner points, z must have the same value at both:
\(z(120, 0) = 120a\)
\(z(60, 30) = 60a+30b\)
\(120a = 60a+30b\)
\(60a = 30b\)
\(b = 2a\)
Check that this common value is the maximum. With b = 2a:
\(z(60, 0) = 60a\)
\(z(40, 20) = 40a+40a = 80a\)
\(z(120, 0) = z(60, 30) = 120a\)
Since a > 0, 120a is the largest. So the condition is \(2a = b\).