Math 209 - Assignment 2 Answer Key - Course Hero Page 4
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1
2
3
4
5Solution:
2
2
2
Let f (x, y, z) = x
+ 2y
+ 3z
. The normal vector of the plane 3x
2y + 3z = 1 is
3, 2, 3 . The normal vector for tangent plane at the point (x
, y
, z
) on the ellipsoid
0
0
0
is
f (x
, y
, z
) = 2x
, 4y
, 6z
. Since the tangent plane is parallel to the given plane,
0
0
0
0
0
0
f (x
, y
, z
) = 2x
, 4y
, 6z
= c 3, 2, 3 or x
, 2y
, 3z
= k 3, 2, 3 . Thus x
=
0
0
0
0
0
0
0
0
0
0
2
2
2
2
14
3k, y
=
k and z
= k. But x
+ 2y
+ 3z
= 1 or (9 + 2 + 3)k
= 1,so k =
and
0
0
0
0
0
14
14
14
14
there are two such point (
,
,
) .
14
14
14
8. Find the local maximum and minimum values and saddle point(s) of the function
2
3
2
2
f (x, y) = 3x
y + y
3x
3y
+ 2.
Solution:
The first order partial derivatives are
2
2
f
= 6xy
6x,
f
= 3x
+ 3y
6y .
x
y
So to find the critical points we need to solve the equations f
= 0 and f
= 0. f
= 0
x
y
x
implies x = 0 or y = 1 and when x = 0, f
= 0 implies y = 0 or y = 2; when y = 1, f
= 0
y
y
2
implies x
= 1 or x =
1. Thus the critical points are (0, 0), (0, 2), ( 1, 1).
2
2
2
Now f
= 6y
6, f
= 6y
6 and f
= 6x. So D = f
f
f
= (6y
6)
36x
.
xx
yy
xy
xx
yy
xy
Critical point Value of f
f
D
Conclusion
xx
(0, 0)
2
-6
36
local maximum
(0, 2)
-2
6
36
local minimum
(1, 1)
0
0
-36
saddle point
( 1, 1)
0
0
-36
saddle point
2
2
9. Find the points on surface x
y
z = 1 that are closest to the origin.
Solution:
The distance from any point (x, y, z) to the origin is
2
2
2
d =
x
+ y
+ z
1
2
2
but if (x, y, z) lies on the surface x
y
z = 1, then z =
and so we have
2
2
x
y
2
2
4
4
d =
x
+ y
+ x
y
.
We can minimize d by minimizing the simpler expression
2
2
2
4
4
d
= x
+ y
+ x
y
= f (x, y) .
4
4
4
4
f
= 2x
, f
= 2y
, so the critical points occur when 2x =
and 2y =
x
y
5
4
4
5
5
4
4
5
x
y
x
y
x
y
x
y
1
1
6
4
4
6
2
2
10
or x
y
= x
y
so, x
= y
and x
= 2
x =
2
, y =
2
. The four critical points
10
10
1
1
1
1
( 2
, 2
). Thus the points on the surface closes to origin are ( 2
, 2
). There is
10
10
10
10
no maximum since the surface is infinite in extent.
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