Find and classify all critical points for the function f(x, y) = x³ - 12xy² + y³ + 45y. A) (2, 1) and (-2, -1) are saddle points B) (2, 1), (2, -1), (-2, 1), and (-2, -1) are saddle points C) (2, 1) is a local maximum and (-2, -1) is a local minimum D) (2, -1) and (-2, 1) are saddle points E) There are no critical points.

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Quizplus · Accepted Answer

The critical points are found by solving the system of equations $$f_x = 3x^2 - 12y = 0$$ $$f_y = -24xy + 3y^2 + 45 = 0$$ Solving this system gives the critical points (2, 1) and (-2, -1). To classify these critical points, we compute the discriminant $$D = f_{xx}f_{yy} - (f_{xy})^2 = (6x)(-6y) - (-12y)^2 = -36xy + 144y^2$$ At (2, 1), D = -36(2)(1) + 144(1)^2 = 108 > 0 and $$f_{xx} = 6x = 6(2) = 12 > 0$$ so (2, 1) is a local maximum. At (-2, -1), D = -36(-2)(-1) + 144(-1)^2 = 108 > 0 and $$f_{xx} = 6x = 6(-2) = -12 < 0$$ so (-2, -1) is a local minimum.

Find and Classify All Critical Points for the Function F(x