4. Find the absolute maximum and minimum values of f(x, y) = xy^2 on the domain

Question

4. Find the absolute maximum and minimum values of f(x, y) = xy^2 on the domain D ={(x, y)|x ? 0, y ? 0, x2 + y
2 ? 3}.

5. Find the points on the cone z^2 = x^2 + y^2 that are closest to the point (4, 2, 0).

6. For functions of one variable it is impossible for a continuous function to have two local maxima
and no local minimum. But for functions of two variables such a situation is possible. Show
that the function f(x, y) = ?(x^2 ? 1)^2 ? (x^2y ? x ?1)^2 has only two critical points and has local maxima at both of them. Then use a 3D plotting program to graph the function (be careful on what viewing window you choose). What do you see happening in the graph that makes this possible? Is it something that could occur in 2D?

Explanation / Answer

find your solution in the attached links.

http://www.math.purdue.edu/~jchavezc/QuizzesMA261f12_files/SolutionsQ6.pdf

https://people.math.osu.edu/joecken.1/documents/m254/hw4s.pdf

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