The corner points of the feasible region determined by x+y ≤ 8, 2 x+y ≥ 8, x ≥ 0, y ≥ 0 are A(0,8), B(4,0) and C(8,0). If the objective function Z=a…

Let's calculate the value of $Z$Z at each corner point:

At point $A(0,8)$A(0,8): $Z_A = a(0) + b(8) = 8b$ZA​=a(0)+b(8)=8b

At point $B(4,0)$B(4,0): $Z_B = a(4) + b(0) = 4a$ZB​=a(4)+b(0)=4a

At point $C(8,0)$C(8,0): $Z_C = a(8) + b(0) = 8a$ZC​=a(8)+b(0)=8a

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For a maximum to occur on segment $AB$AB (not just at endpoints), the function must increase from $C$C to $B$B and from $B$B to $A$A.

For $Z$Z to increase from $C$C to $B$B: $Z_B > Z_C$ZB​>ZC​ which means $4a > 8a$4a>8a, so $a < 0$a<0

For $Z$Z to increase from $B$B to $A$A: $Z_A > Z_B$ZA​>ZB​ which means $8b > 4a$8b>4a, so $2b > a$2b>a

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For the maximum to occur exactly on segment $AB$AB, the level curves of $Z$Z must be parallel to $AB$AB.

The slope of $AB$AB is:

$$\dfrac{0-8}{4-0} = \dfrac{-8}{4} = -2$$4−00−8​=4−8​=−2

The slope of the level curves $Z=ax+by$Z=ax+by is $-\dfrac{a}{b}$−ba​

For these to be parallel: $-\dfrac{a}{b} = -2$−ba​=−2

Which simplifies to: $a = 2b$a=2b

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The relation between $a$a and $b$b is $a = 2b$a=2b.

We can verify: If $a = 2b$a=2b, then $Z_A = 8b$ZA​=8b and $Z_B = 4a = 4(2b) = 8b$ZB​=4a=4(2b)=8b, confirming equal values at both points.