An objective function \(Z = ax + by\) is maximum at points (8, 2) and (4, 6). If \(a \ge 0\) and \(b \ge 0\) and ab = 25, then the maximum value of the function is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 50
Since Z is maximum at both points, it has the same value at (8, 2) and (4, 6):
\(8a + 2b = 4a + 6b\)
\(4a = 4b\)
\(a = b\)
Using ab = 25:
\(a^2 = 25\)
Since \(a \ge 0\), a = 5 and so b = 5. The maximum value is
\(Z(8, 2) = 8(5) + 2(5) = 50\)
As a check, \(Z(4, 6) = 4(5) + 6(5) = 50\).