For constants ?, ? and ?, let ?(?,?) = ??2 + ??? + ??2 and ? = 4?? − ?2. Show without using the second derivative test that (a) If ?,? > 0, then ? has a local minimum at (0,0). (b) If ? > 0 and ? < 0, then ? has a local maximum at (0,0). (c) If ? < 0, then (0,0) is a saddle point.
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Please do question 5a and 5b 4. In this problem we analyze the behavior of the polynomial f (x, y) = ax² + bxy + cy? (without using the Second Derivatives Test) by identifying the graph as a paraboloid. (a) By completing the square, show that if a + 0, then b 2 4ac - 62 f(x, y) = ax² + bxy + cy? = a [( 2 + Y + 2a 4a2 (b) Let D = 4ac – 62....
9. (10 pts.) The following function has three critical points (0,0), (1,1), (-1,-1). Use the second derivative test to determine if each point is a local maximum, local minimum, or saddle point. S(1,y) = 1 + y - 4xy +1.
2. For the two-argument function defined below: f(x,y) = 2x2 – 8xy + 5y + 3y2 (a) Find fx = and fex = . (5 marks) (b) Find fy = and fyy (5 marks) (c) Determine the critical point(s) of the f(x,y). (8 marks) (d) Find fxy (3 marks) (e) Determine each of the critical point(s) in the above (c) whether is a local minimum, local maximum or saddle point by using second partial derivative test. (4 marks)
This two-variable function has exactly two critical points, including the origin: 43 13 f(x,y) = xy - Q17.1 FR 5 Part (a) 4 Points Find the second critical point, other than (0,0). Please select file(s) Select file(s) Q17.2 FR 5 Part (b) 4 Points Classify the critical point (0,0) as a local min, local max, or saddle point, using the (multivariate) second-derivative test.
Let f(x,y) = 4 + x² + y² – 3xy f has critical points at 10,0) and (1,1) use the second derivative test to classify these points as local min, local max, or saddle point
Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. 1. f(x, y) = 4.cy - 24 – 44
Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle point. f(x y) = (x2 - 25)2 + (02-81 Of(0, 0) - 7186 local maximum; f(0, 9) - 625, saddle point; f(0-9) - 625, saddle point: f(5,0) = 7186 saddle point: f(5,9) - 0, local minimum;f(5, -9) - 0, local minimum:f(-5,0) = 6561, saddle point; 6(-5, 9) - 0, local minimum: f(-5, -9) - 0, local minimum Of(0,0) - 7186,...
Cal 4 , ) and use this to 6. Let f(x,y) = x2 + y2 + 2x + y. (a) Find all critical points of f in the disk {(x,y) : x2 + y2 < 4). Use the second derivative test to determine if these points correspond to a local maximum, local minimum, or saddle point. (b) Use Lagrange multipliers to find the absolute maximum/minimum values of f(x, y) on the circle a2 +y -4, as well as the points...
Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. .f(x, y) = x²y2
Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. f(x, y) = x2 + 4xy + y21