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# C false 00 lies is feasible region than there are

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c) False (0,0) lies is feasible region than there are chances for it to be an optimal solution. If we look at example z= -x 1 -x 2 We get below graph with (0,0) as optimal solution.

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3.2-3) a) Requirement of friend Total available Friend 1 Friend 2 Investment 5000 4000 6000 Full time hours 400 500 600 Profit \$4500 \$4500 b) Let x 1 = Fraction of full partnership for friend 1. x 2 = Fraction of full partnership for friend 2. z= 4500x 1 + 4500x 2 Subject to x 1 ≤ 1, x 2 ≤ 1 5000x 1 + 4000x 2 ≤ 6000 400x 1 + 500x 2 600

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c) 3.2-6) Given: -x 1 + 2x 2 ≤ 50 -2x 1 + x 2 ≤ 50 Where x 1 ≥ 0, x 2 ≥ 0 a) Below graph demonstrates that the feasible region is unbounded.
b) Maximize: z = -x 1 + x 2 Subject to - x 1 + 2x 2 ≤ 50 -2x 1 + x 2 ≤ 50 Where x 1 ≥ 0, x 2 ≥ 0

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Optimal Solution: z = 25 with x 1 =0 & x 2 = 25 c) Maximize: z = x 1 - x 2 Subject to - x 1 + 2x 2 ≤ 50 -2x 1 + x 2 ≤ 50 Where x 1 ≥ 0, x 2 ≥ 0
Below graph shows that this objective function has no optimal solution. If we consider any random value i.e. z= 10 Z= x 1 - x 2 = 10 Now if x 1 = 0, x 2 = -10 x 2 = 0, x 1 = 10 We will be a line slanting towards right (/) and thus the region is unbounded we will not get an optimal solution.

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d) For the above example where there is no optimal solution, there is
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