The corner points of the bounded feasible region determined by the system of linear inequalities are (0, 0), (2, 4), (0, 5) and (4, 0). If the maximum value of \(Z = ax + by\), where \(a, b \gt 0\) occurs at both (2, 4) and (4, 0), then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(a = 2b\)
Since the maximum occurs at both (2, 4) and (4, 0), Z has the same value at these two corner points:
\(Z(2, 4) = 2a + 4b,\quad Z(4, 0) = 4a\)
\(2a + 4b = 4a\)
\(4b = 2a,\quad a = 2b\)
Check with a = 2b: Z is 8b at both (2, 4) and (4, 0), while the other corners give smaller values, since b > 0:
\(Z(0, 5) = 5b,\quad Z(0, 0) = 0\)