f(x,y)=3x-x^3-2y^2+y^4 A. Find all critical points of f. B. Does f have an absol

Question

f(x,y)=3x-x^3-2y^2+y^4 A. Find all critical points of f. B. Does f have an absolute maximum or minimum? If yes, find them and explain how you know they are absolute extrema. If no, explain why there are no absolute extrema. C. Suppose the domain of f is {(x,y): -3 is less than or equal to x is less than or equal to 3 and -2 is less than y is less than or equal to 2}. Does f have an absolute maximum or minimum? If yes, find them and explain how you know they are absolute extrema. If no, explain why there are no absolute extrema

Explanation / Answer

Set these equal to 0: 3x^2 - 6y = 0 -6x + 2y = 0 ==> y = 3x. Substituting yields 3x^2 - 6(3x) = 0 ==> x = 0, 6. Hence, the critical points are (x, y) = (0, 0), (6, 18). ----------------------------- Classify these with the second derivative test. f_xx = 6x, f_yy = -2, f_xy = -6 ==> D = (f_xx)(f_yy) - (f_xy)^2 = -12x - 36. Since D(0, 0) = -36 < 0, we have a saddle point at (0, 0). Since D(6, 18) = -108 < 0, we have another saddle point at (6, 18). I hope this helps!

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