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(x 2 1) 2 (x 2 y 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

Question: 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

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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