Science:Math Exam Resources/Courses/MATH105/April 2016/Question 01 (d)

MATH105 April 2016

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Question 01 (d)
Determine whether the following statement is true: The function $\mathbf {f(x,y)}$ could have both an absolute maximum and an absolute minimum at two different points that are not critical points.

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
Consider the statement on a bounded 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. Suppose that $f(x,y)=x+y$ and that the domain of $f$ is $[0,1]^{2}$ (i.e., the set of points $(x,y)$ satisfying $0\leq x\leq 1$ and $0\leq y\leq 1$). Since $0\leq x\leq 1$ and $0\leq y\leq 1$, clearly $f$ has an absolute minimum of $0+0=0$ at $(0,0)$ and an absolute maximum of $1+1=2$ at $(1,1)$. However, neither $(0,0)$ nor $(1,1)$ is a critical point of $f$ because $\nabla f=(\partial _{x}f,\partial _{y}f)=(1,1)$ (recall that a critical point of $f$ is a point $(x,y)$ at which $\nabla f(x,y)=(0,0)$). Therefore, the statement is true.