UT M 408C - HW 14.7-solutions-1 (11 pages)

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HW 14.7-solutions-1



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HW 14.7-solutions-1

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Pages:
11
School:
University of Texas at Austin
Course:
M 408c - Differential and Integral Calculus

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lewis scl876 HW 14 7 radin 55315 This print out should have 17 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 10 0 points Suppose 1 1 is a critical point of a function f having continuous second derivatives such that 1 at 1 1 002 10 0 points In the contour map below identify the points P Q and R as local minima local maxima or neither 3 2 fxx 1 1 5 fxy 1 1 2 1 0 1 fyy 1 1 5 2 P Q Which of the following properties does f have at 1 1 0 R 2 1 a saddle point correct 1 0 1 2 a local maximum 2 3 3 a local minimum Explanation Since 1 1 is a critical point the Second Derivative test ensures that f will have A local minimum at R B local maximum at P C neither local max nor local min at Q i a local minimum at 1 1 if fxx 1 1 0 fyy 1 1 0 fxx 1 1 fyy 1 1 fxy 1 1 2 1 A only 2 B only 3 C only correct ii a local maximum at 1 1 if fxx 1 1 0 fyy 1 1 0 fxx 1 1 fyy 1 1 fxy 1 1 2 4 A and C only 5 A and B only 6 all of them iii a saddle point at 1 1 if 7 B and C only 2 fxx 1 1 fyy 1 1 fxy 1 1 From the given values of the second derivatives of f at 1 1 it thus follows that f has a saddle point 8 none of them Explanation A FALSE the contours near R are closed curves enclosing R and the contours increase in value as we approch R So the surface has a local maximum at R not a local minimum lewis scl876 HW 14 7 radin 55315 B FALSE the contours near P are closed curves enclosing P and the contours decrease in value as we approch P So the surface has a local minimum at P not a local maximum C TRUE the point Q lies on the 0 contour and in this contour divides the region near Q into two regions In one region the contours have values increasing to 0 while in the other the contours have values decreasing to 0 So the surface has neither a local maximum nor a local minimum at Q keywords contour map local extrema True False 003 part 1 of 3 10 0 points If f is defined by 8 f x y x3 x2 4xy 2y 2 y 3 locate the critical points of f 1 3 1 0



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