# Classification local minimum local maxi mum saddle

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) Classification: ? (local minimum, local maxi- mum, saddle point, cannot be determined) 14. (1 pt) The diagram at the left represents a collection of level sets for a certain function, where the outer-most level is at the lowest height. Point A is ? Point B is ? Point C is ? Point D is ? Point E is ? 15. (1 pt) Consider the function f ( x , y ) = ( 6 x - x 2 )( 8 y - y 2 ) . Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank. f x = f y = f xx = f xy = f yy = There are several critical points to be listed. List them lexi- cograhically, that is in ascending order by x-coordinates, and for equal x-coordinates in ascending order by y-coordinates (e.g., (1,1), (2, -1), (2, 3) is a correct order) The critical point with the smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) 16. (1 pt) 3

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The diagram at the left represents a collection of level sets for a certain function, where the outer-most level is at the lowest height. Point A is ? Point B is ? Point C is ? Point D is ? 17. (1 pt) Find the maximum and minimum values of the function f ( x , y ) = 2 x 2 + 3 y 2 - 4 x - 5 on the domain x 2 + y 2 36. As usual, ignore unneeded answer blanks, and list points lexi- cographically. Maximum value is , occuring at ( , ), and ( , ). Minimum value is , occuring at ( , ) and ( , ). 18. (1 pt) Find the minimum and maximum values of the function sub- ject to the given constraint f ( x , y ) = 3 x 2 + 2 y 2 , x + 2 y = 5 Enter DNE if such a value does not exist. f min = f max = Solution: Solution: We find the extreme values of f ( x , y ) = 3 x 2 + 2 y 2 under the con- straint g ( x , y ) = x + 2 y - 5 = 0 We write out the Lagrange Equations. The gradients of f and g are f = h 6 x , 4 y i and g = h 1 , 2 i .

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