Since both points (8,2) and (4,6) give the same maximum value of Z, we can write:
Zmax=a(8)+b(2)=a(4)+b(6)
8a+2b=4a+6b
4a=4b
a=b
Using the constraint ab=25 and knowing that a=b:
a2=25
a=5 (since a≥0)
Thus, b=5 as well.
Calculating the maximum value:
Zmax=a(8)+b(2)=5(8)+5(2)=40+10=50
We can verify using the second point: Zmax=5(4)+5(6)=20+30=50
Therefore, the maximum value of the function is 50.