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Unformatted text preview: moseley (cmm3869) – HW14 – Gilbert – (56380) 1 This printout should have 9 questions. Multiplechoice 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 112321 1 2 Q P R A. local maximum at Q , B. local minimum at P , C. local minimum at R . 1. C only 2. A and C only 3. A and B only 4. all of them 5. A only 6. B and C only 7. B only correct 8. none 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. 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 . C. 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. 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 (0 , 0), saddle point at (1 , − 1) 2. local max at (1 , − 1), local min at (0 , 0) 3. local min at (1 , − 1), saddle point at (0 , 0) correct 4. local max at (1 , − 1), saddle point at (0 , 0) 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 = 3 x 2 + 3 y = 0 , i.e., y = − x 2 , moseley (cmm3869) – HW14 – Gilbert – (56380) 2 while f y = 0 when ∂f ∂y = − 3 y 2 + 3 x = 0 , i.e., x = y 2 ....
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 Spring '07
 Sadler
 Critical Point, Multivariable Calculus, Fermat's theorem, local minimum

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