suppose f (x,y) = x^2+y^2-4x-6y+1 B.If there is a local minimum, what is the val
ID: 3188909 • Letter: S
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suppose f (x,y) = x^2+y^2-4x-6y+1 B.If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N. (C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N. (D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. (E) What is the maximum value of f on R^2 ? If there is none, type N. (F) What is the minimum value of f on R^2? If there is none, type N.Explanation / Answer
Suppose f(x,y)=x^2+y^2-4x-6y+5 (A) How many critical points does f have in R^2? Answer: 1 (B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N. Answer: can't figure out. please help (C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N. Answer: N (D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. Answer: N (E) What is the maximum value of f on R^2? If there is none, type N. Answer: N (F) What is the minimum value of f on R^2? If there is none, type N. Answer: can't figure out. please help Given: f(x,y) = x^2 + y^2 - 4x - 6y + 5 First, find the critical points. f_x = 2x - 4, and f_y = 2y - 6. Setting these equal to 0 yields the critical point (x,y) = (2, 3). -------------------------------- Using the Second Derivative Test: f_xx = 2, f_yy = 2, f_xy = 0 ==> D = (f_xx)(f_yy) - (f_xy)^2 = 2 * 2 - 0 = 4. Since D(2, 3) = 4 > 0 and f_xx(2, 3) > 0, we have a local minimum at (2, 3). -------------------------------- In fact, this is a global minimum (via completing the square): f(x,y) = x^2 + y^2 - 4x - 6y + 5 ........= (x^2 - 4x) + (y^2 - 6y) + 5 ........= (x^2 - 4x + 4) + (y^2 - 6y + 9) + 5 - 4 - 9 ........= (x - 2)^2 + (y - 3)^2 - 8 We have a global minimum of -8 where x - 2 = 0 and y - 3 = 0 (x, y) = (2, 3).Related Questions
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