Maximise Z = 3x + 6y subject to x + 2y ≤ 8, x ≤ 6, x ≥ 0, y ≥ 0. How many optimal solutions are there?

Question 41

MediumOriginal question

Maximise $Z = 3x + 6y$ subject to $x + 2y \le 8, x \le 6, x \ge 0, y \ge 0$ . How many optimal solutions are there?

Exactly oneCorrect answerYour answerExactly twoCorrect answerYour answerNoneCorrect answerYour answerInfinitely manyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Corners: $(0, 0)$ , $(0, 4)$ , $(6, 0)$ , $(6, 1)$ ; $Z$ is $0$ , $24$ , $18$ , $24$ .
The maximum, 24, comes at both $(0, 4)$ and $(6, 1)$ , because $Z = 3(x + 2y)$ is parallel to the edge $x + 2y = 8$ ; every point of that edge is optimal.

Worth noting

'Exactly two' counts only the corners, but every point between them on the edge gives 24 too.

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