If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum But this is not true for functions of two variables Show that the function f (x, y) 3xe y x 3 e 3y has exactly one critical point, and that f has a local maximum there that is not an absolute maximum Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible

Question: If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an

If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum. But this is not true for functions of two variables. Show that the function
f (x, y) = 3xe^y - x³ - e^3y
has exactly one critical point, and that f has a local maximum there that is not an absolute maximum. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

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