Question

Find the critical points of the following function. Use the Second Derivative Test to determine if possible) whether each critical point corresponds to a local maximum, local minimum or saddle point. Contem your results with a graphing utility f(x,y) = x + xy-2) + 4y - 12 What are the critical points? (Type an ordered pair Use a comma to separate answers as needed.) Use the Second Derivative Test to find the local maxima. Select the correct choice below and, if necessary in the answer box within your choice O A The best shows that there is/are local maxima at (Type an ordered pair. Use a comme to separate answers as needed) OB. The best does not reveal any local maxima and there are no critical points for which the test is inconclusive, so there are no local macima OC. The best does not reveal any local maxima, but there is at least one critical point for which the latis inconclusive Use the Second Derivative Test to find the local minima. Select the correct choice below and necessary, in the ant box within your choice O A The test shows that there is are local minima at contre 2

Answer 1

function is a w.r. t. x and andy ee. Given fix,y) = x + x²(y-2) + ulys Since this is function of two variables So to find critical points, we find partial derivatives 2* and af af and set them zero. ay af 4x² + 2x(y-2) x + 819-1 at =0 and at zo ay 4x3 + 2x 19-2) = 0 2x (2x + and (y-2)) = 0 x² + 8 819-1=0 8y-D .ax at ay Now ax Th 9

or De = 801-8 2 08 2x² + 2 X8 (1- + =0 From ①, either y=1 (y-2) = put value of xes in above equation 8(1-7) -15 Using x² 9.-2) = 16-167 + 8 + 14 =0 14 15 8X 8x11-1945) M = 8X ) 15 x = + Ti 8 15 So critical points are (0,1), TY 15 and 8 17 2 15 To check their local and global extremas we find second derivatives. 2²4 12x²+ (y-2) 2+ = sxzy 12xot (1-2) =- 2x2 2²f = 8 aya At (0, 1) A = 2²7 = are ² (0,1) B = 2²t axay 160,1) c = 22t aye (0,1) 2X0 =0 leon 8 com

(1) At U is - Now AC-B²= -8-0=-8 co So (0,1) is a saddle point. 8 14 V 15 A= 2² 12x 8 2x2 5 15 96- 16 80 15 15 B= 27 8 2 V 15 ۷) 8. G-2) - axay c= 227 aya : Aso and AC-B²= -uk do 80x8 15 15 so 사 14 8 15 1) is a point minima 15 here. ich) At Eli 5) A = 227 axe² B = dxay 8 14 15 15. 12x 8 +/14 -2 15 15 2X 2 2 22f - 80 -15 =-42 c= 227 8 aya Aso and AC-B - So. 80x8 tur2 15 is also a 8 15 15 15 minima point of here
option 2nd is correct here.