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

Since both points $(8,2)$ and $(4,6)$ give the same maximum value of $Z$, we can write:

$Z_{max} = a(8) + b(2) = a(4) + b(6)$

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

$4a = 4b$

$a = b$

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Using the constraint $ab = 25$ and knowing that $a = b$:

$a^2 = 25$

$a = 5$ (since $a \geq 0$)

Thus, $b = 5$ as well.

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Calculating the maximum value:

$Z_{max} = a(8) + b(2) = 5(8) + 5(2) = 40 + 10 = 50$

We can verify using the second point: $Z_{max} = 5(4) + 5(6) = 20 + 30 = 50$

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