For functions of one variable it is impossible for a continuous function to have two local maxima
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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 functions exist. Show that the function
f(x, y) = 2(x2 - 1)2 - (x2y - x - 1)2
has only two critical points, but has local maxima at both of them. 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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