Define g(x, y) = 2xy(x + y)/(x² + y²) for (x, y) ≠ 0 and g(0, 0) = 0. In this exercise, we show that g(x, y) is continuous at (0, 0) and that g_x(0, 0) and g_y(0, 0) exist, but g(x, y) is not differentiable at (0, 0).

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(a) Show using polar coordinates that g(x, y) is continuous at (0, 0). (b) Use the limit definitions to show that gx (0, 0) and gy(0, 0) exist and that both are equal to zero. (c) Show that the linearization of g(x, y) at (0, 0) is L(x, y) = 0. g(h, h) h0 h (d) Show that if g(x, y) were differentiable at (0, 0), we would have lim = 0. Then observe that this is not the case because g(h, h) = 2h. This shows that g(x, y) is not differentiable at (0,0) and hence not differentiable at (0, 0).