Math 209 - Assignment 2 Answer Key - Course Hero Page 5
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1
2
3
4
52
2
10. Find the extreme values of f (x, y) = 2x
+ 3y
4x
5 on the region
2
2
D = (x, y) x
+ y
16 .
Solution:
We first need to find the critical points. These occur when
f
= 4x
4 = 0,
f
= 6y = 0
x
y
2
2
so the only critical point of f is (1, 0) and it lies in the region x
+ y
16.
2
2
2
2
On the circle x
+ y
= 16, we have y
= 16
x
and
2
2
2
2
g(x) = f (x, 16
x
) = 2x
+ 3(16
x
)
4x
5 =
x
4x + 43 .
g (x) = 0
2x
4 = 0
x =
2
2
2
y
= 16
x
= 16
4 = 12
y =
2 3 .
Now f (1, 0) =
7 and f ( 2, 2 3) = 47. Thus the maximum value of f (x, y) on the disc
2
2
x
+ y
16 is f ( 2, 2 3) = 47, and the minimum value is f (1, 0) =
7.
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