Science:Math Exam Resources/Courses/MATH100 B/December 2023/Question 24

MATH100 B December 2023

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Question 24
Find the absolute maximum of the function $$f(x,y)=2x-3y$$ on the rectangular domain ${\textstyle 0\leq x\leq 2}$ and ${\textstyle 0\leq y\leq 1}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that on a closed domain, the absolute maximum (or absolute minimum) is found on the boundary of the domain.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We know the absolute maximum is found on the boundary of the domain. We can also see from the function ${\textstyle f}$ and its domain that ${\textstyle x}$ has a positive coefficient, ${\textstyle y}$ has a negative coefficient, and the intervals where ${\textstyle x}$ and ${\textstyle y}$ can exist are both positive. From this we can deduct that ${\textstyle f(x,y)}$ will be maximised when ${\textstyle x}$ is at the upper bound of its domain, and ${\textstyle y}$ is at the lower bound of its domain. This is $$f(2,0)=2(2)-3(0)=4.$$