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hw 14 - rhodes(ajr2283 HW14 Gilbert(56380 This print-out...

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rhodes (ajr2283) – HW14 – Gilbert – (56380) 1 This print-out should have 9 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 0 -3 -2 -1 0 1 2 Q P R A. local maximum at Q , B. local minimum at R , C. local maximum at P . 1. all of them 2. B only 3. A only 4. C only 5. none of them correct 6. A and B only 7. B and C only 8. A and C only 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. 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. 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 + 3 xy 1 . 1. local min at (1 , 1), saddle point at (0 , 0) correct 2. local max at (1 , 1), local min at (0 , 0) 3. local max at (1 , 1), saddle point at (0 , 0) 4. local max at (0 , 0), saddle point at (1 , 1) 5. local min 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 .

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hw 14 - rhodes(ajr2283 HW14 Gilbert(56380 This print-out...

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