7. Let f(x,y) = 2y2 – 15y + x2y – 2xy. At which of the following critical points does f have a relative maximum?
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I. (-3,0)
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II. (1,4) |
III. (5,0) |
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A. None |
B. I and II only |
C. I and III only |
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D. II and III only |
E. The correct answer is not given by A,B,C, or D |
Solution: The answer is A
Recall the second derivative test: Let (a,b) be a critical point of f(x,y). Set
Then
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If D(a,b) < 0 then f(a,b) is neither a local min nor max |
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If D(a,b) > 0 and fxx(a,b) > 0 then f(a,b) is a local min |
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If D(a,b) > 0 and fxx(a,b) < 0 then f(a,b) is a local max |
Now
so
and
At (-3,0) we have D(-3,0)-4(-4)2 <0, so (-3,0) is a saddle point.
At (1, 4) we have D(1,4) = 32 > 0 and fxx(1,4) = 8 > 0, so there is a local min at (1,4).
At (5,0), D(5,0) < 0, so (5,0) is a saddle.