If the objective function \(z = 4x + 3y\) has maximum value on a line joining points \((3, a)\) and \((b, 2)\) where \(a \gt 0, b \gt 0\) such that \(a - b = 2\), then the maximum value of \(z\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 54
If the maximum is reached on the whole segment joining \((3, a)\) and \((b, 2)\), then z takes the same value at both ends of the segment:
\(4(3) + 3a = 4b + 3(2)\)
\(12 + 3a = 4b + 6\)
\(4b - 3a = 6\)
Putting \(a = b + 2\):
\(4b - 3(b + 2) = 6\)
\(b = 12,\quad a = 14\)
So the maximum value is
\(z(3, 14) = 12 + 42 = 54\)
\(z(12, 2) = 48 + 6 = 54\)