Question

An objective function $Z = ax + by$ is maximum at points $(8, 2)$ and $(4, 6)$. If $a \geq 0$ and $b \geq 0$ and $ab = 25$, then the maximum value of the function is:

• A. $60$
• B. $50$
• C. $40$
• D. $80$

Answer 1

Answer: (B)

$50$

Answer 2

The given function $Z = ax + by$ attains its maximum value at points $(8, 2)$ and $(4, 6)$. At these points:

$Z_1 = 8a + 2b \quad \text{and} \quad Z_2 = 4a + 6b$

Since both points yield the same maximum value:

$8a + 2b = 4a + 6b$

Simplify the equation:

$8a - 4a = 6b - 2b$

$4a = 4b$

$a = b$

Using the condition $ab = 25$:

$a \cdot b = 25 \quad \text{and} \quad a = b$

$a^2 = 25 \implies a = 5 \quad \text{and} \quad b = 5$

Substitute $a = 5$ and $b = 5$ into $Z = ax + by$. At point $(8, 2)$:

$Z = 8a + 2b = 8(5) + 2(5) = 40 + 10 = 50$

Thus, the maximum value of $Z$ is 50.