Quiz5-3_15

# Quiz5-3_15 - Name You have fteen minutes to complete the...

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Name: You have fifteen minutes to complete the quiz. If anything seems unclear, please ask. 1. The function f ( x, y ) = x 3 - 3 x 2 + y 2 has two critical points. (a) Find both critical points. (3 points) (b) Classify the critical points (local minimum, local maximum, saddle, or not enough information). (3 points) (c) Using this information, which of the pictures overleaf is the graph of f ? (1 point) 2. Find the points, and values, where the function f ( x, y ) = 2 - x 2 - y 2 attains a global minimum and global maximum on the closed unit disk { ( x, y ) R 2 | x 2 + y 2 1 } . (3 points) 1

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(a) (b) (c) (d) 2
Solutions If you have any questions and / or think you’ve spotted a mistake / typo, please get in touch with me. 1. (a) The gradient vector is f ( x, y ) = h 3 x 2 - 6 x, 2 y i ; setting this equal to zero gives 3 x 2 - 6 x = 0 x ( x - 2) = 0 x = 0 , 2 and 2 y = 0 y = 0 , so the critical points are at (0 , 0) and (2 , 0). (b) We have that D ( x, y ) = f xx ( x, y ) f yy ( x, y ) - f xy ( x, y ) 2 = (6 x - 6)(2) = 12
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