Thomas Calculus 13th [Solutions]

1046 Chapter 14 Partial Derivatives
temperatures are T (0, 1, 0) 0, T ( 1, 0, 0) 0, and
1 1 2
T , , 50. Therefore 50 is the maximum
2 2 2
temperature at
1 1 2
, , and
2 2 2
1 1 2
, , ; 50 is the minimum temperature at
2 2 2
1 1 2
, , and
2 2 2
1 1 2
, , .
2 2 2
31. (a) If we replace x by 2x and y by 2y in the formula for P,
P changes by a factor of
(b) For the given information, the production function is
G( x, y) 250x 400y 100,000 0. P G implies
Eliminating
yields
1/4
x
90 250
y
x
y
3/4
x
30 400
y
24
5
or 5x
24 y . Now we solve the system
5x
24y
0
250x
400y
100,000
and find x 300, y 62.5. P(300, 62.5) 24,322 units.
1
2 2 2.
3/4 1/4
P( x, y) 120x y and the constraint is
32. We can omit k since it affects the value but not the location of the maximum. We then have
1
P( x, y)
x y and G( x, y) c1x c2
y B 0. P G implies
(1 )
x c
Eliminating yields
2
y c
1
1
x
y
x
y
.
1
1
c
1
1
c
2
Now we solve the system
and find
x
1 2
c (1 ) x c y
1 2
c x c y B
1 2
B (1 ) B
, y .
c c
33. U ( y 2) i xj and g 2i j so that U g ( y 2) i xj (2 i j ) y 2 2 and x
y 2 2x y 2x 2 2 x (2x 2) 30 x 8 and y 14. Therefore U (8, 14) $128 is the
maximum value of U under the constraint.
Copyright
2014 Pearson Education, Inc.