Question 1
Find the maximum and minimum values of f(x, y) = (x - y) e
-(x2 + y2).
◦ maximum e
1/2, minimum - e
1/2◦ maximum
![](data:image/png;base64, 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)
e
-1/2, minimum -
![](data:image/png;base64, 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)
e
-1/2◦ maximum 2 e
-1/2, minimum 0
◦ maximum e
-1/2, minimum 0
◦ maximum e
-1/2, minimum -e
-1/2Question 2
Find and classify all critical points for the function f(x, y) = 2y
3 + 3y
2 - 12y - x
2 + 2x.
◦ (1, -2) local maximum; (1, 1) local maximum
◦ (-1, 2) saddle point; (1, 1) saddle point
◦ (1, -2) local maximum; (1, 1) saddle point
◦ (1, -2) local minimum; (1, 1) local minimum
◦ (1, -2) local maximum; (1, 1) local minimum