Explanation:

- The problem provides an objective function \( Z = ax + by \) that is maximized at points (8, 2) and (4, 6).

- Given that \( a > 0 \) and \( b > 0 \), with \( ab = 25 \), both points should yield the same maximum value for the function.

- Let's calculate the function at both points:

- For point (8, 2): \( Z = 8a + 2b \)

- For point (4, 6): \( Z = 4a + 6b \)

- Since both expressions are equal, set \( 8a + 2b = 4a + 6b \).

- This simplifies to \( 4a = 4b \), so \( a = b \).

- Given \( ab = 25 \) and \( a = b \), substitute to find \( a^2 = 25 \).

- Thus, \( a = 5 \) and \( b = 5 \).

- Calculating max value: \( Z = 8a + 2b = 8 \cdot 5 + 2 \cdot 5 = 40 + 10 = 50 \).

- Therefore, the maximum value of the function is 50.

- ?? The correct option is 2: 50.

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