# 22_new - HW10 gilbert(55035 This print-out should have 16...

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HW10 gilbert (55035) 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points In the contour map below identify the points P, Q, and R as local minima, local maxima, or neither. 3 2 1 0 -1 -2 P Q 0 R 2 1 0 -1 -2 -3 A. local maximum at Q, B. local minimum at R, C. local minimum at P . 1. A and C only 2. A and B only 3. B and C only 4. C only correct 5. none of them 6. A only 7. B only 8. all of them Explanation: A. FALSE: the point Q lies on the 0- contour and 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 does not have a local minimum at Q. B. 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. C. TRUE: 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 . keywords: contour map, local extrema, True/False, 002 10.0 points Locate and classify all the local extrema of f (x, y) = x 3 y 3 + 3xy 2 . 1. local max at (1, 1), local min at (0, 0) 2. local max at (1, 1), saddle point at (0, 0) 3. local min at (0, 0), saddle point at (1, 1) 4. local min at (1, 1), saddle point at (0, 0) correct 5. local max at (0, 0), saddle point at (1, 1) Explanation: Since f has derivatives everywhere, the crit-ical points occur at the solutions of f (x, y) = f X i + f Y j = 0 . But f X = 0 when f x = 3x 2 + 3y = 0 , i.e., y = x 2 ,

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