Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 10 (a)

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MATH100 C December 2024

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• December 2023 • December 2024 •

Question 10 (a)
Consider the function $f (x, y) = x y$ on the domain $x^{2} + y^{2} \leq 1 .$ Find all critical points in the domain.

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 the definition: A critical point of a function $f (x)$ is a point in its domain where the derivative is zero or undefined.

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Solution
Following the definition of the critical point we compute the following derivatives: $f_{x} = y, and f_{y} = x .$ Both $f_{x}$ , and $f_{y}$ are defined everywhere on the domain, so the only critical point occurs when the derivative is zero, which is at $(0, 0)$ .

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Question 10 (a)

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