An objective function is maximum at points and . If and , then the maximum value of the function is…
An objective function $Z = ax + by$ is maximum at points $(8,10)$ and $(7,12)$. If $a, b \ge 0$ and $ab = 72$, then the maximum value of the function is equal to:
- a84
- b132
- c156
- d176
Answer
Answer
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Correct option: c
(c) 156
Since $Z = ax + by$ attains its maximum at both $(8, 10)$ and $(7, 12)$:
$8a + 10b = 7a + 12b \Rightarrow a = 2b$.
Given $ab = 72$, substituting $a = 2b$ gives $(2b)(b) = 72 \Rightarrow 2b^2 = 72 \Rightarrow b^2 = 36 \Rightarrow b = 6$ (since $b \ge 0$).
Then $a = 2(6) = 12$.
The maximum value is $Z = 8(12) + 10(6) = 96 + 60 = 156$.
From ISC 2027 Specimen Mathematics Paper 1, question 1(xiii).