Hello,

Please help me solve these multivariable local minimum/maximum problems. Please be sure to write out all work in a thorough and detailed manner. I have included a solution key for the questions I have posted at the very bottom. Please be sure to write out answers that exactly correspond to the answers. Thanks in advance.

For reference, I would only like questions 31, 33 and 43 solved with explanation.

and (-1, -1) 29. (0, 0); Second Derivative Test is inconclusive; absolute min of 4 at (0, 0) 31. Local min at (2, 0) 33. Saddle point at (0, 0); local max at and 2 2 ; local min at and 2 2 35. Local min at (-1, 0); local max at (1, 0) 37. Saddle point at (0, 1 ); local min at ( 1 2, 0) 39. Saddle point at (0, 0) 41. Saddle point 43. Height = 32 in, base is 16 in X 16 in; volume is 8192 in23-40. Analyzing critical points Find the critical points of the follow- ing functions. Use the Second Derivative Test to determine ( if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclu sive, determine the behavior of the function at the critical points. 23. f(x, y) = -4x2 + 8y2 - 3 24. f (x, y ) = x4 + 14 - 4x - 32y + 10 25. f(x, y) = 4+ 2x2 + 3y2 26. f(x, y) = xye x-y 27. f(x, y) = x4 + 2y2 - 4xy 28. f(x, y) = (4x - 1)2 + (2y + 4)2+ 1 29. f(x, y ) = 4 + x4+ 3y4 30. f(x, y) = x4y2 31. f(x, y) = Vx2 + y2 - 4x + 5 32. f(x, y) = tan xy 33. f(x, y) = 2xye-72-yz 34. f(x, y) = x2 + xy2 - 2x+ 143. Shipping regulations A shipping company handles rectangular boxes provided the sum of the height and the girth of the box does not exceed 96 in. (The girth is the perimeter of the smallest side 01 the box.) Find the dimensions of the box that meets this condition and has the largest volume