Thomas Calculus 13th [Solutions]

1034 Chapter 14 Partial Derivatives
2 2
2x 2y 12x 8y 52 Dx
( x, y) 4x 12 0 and Dy
( x, y) 4y 8 0 critical point is ( 3, 2)
z
13;
2
Dxx ( 3, 2) 4, Dyy ( 3, 2) 4, Dxy ( 3, 2) 0 DxxDyy D xy 16 0 and D xx 0 local
minimum of d( 3, 2, 13) 26
57. V ( x, y, z) (2 x)(2 y)(2 z) 8 xyz;
and
2 2 2 2 2 2 2
x y z 4 z 4 x y V ( x, y) 8xy 4 x y , x 0
2 3
32 y 16x y 8y
32x 16xy 8x
y 0 Vx
( x, y ) 0 and V
2 2
y ( x, y ) 0 critical points are (0, 0),
2 2
4 x y
4 x y
2 , 2 , 2 , 2 , 2 , 2 , and 2 , 2 . Only (0, 0) and 2 , 2 satisfy x 0 and y 0
3 3 3 3 3 3
3 3
3 3
V (0, 0) 0 and V 2 , 2 64 ; On
3 3 3 3
V (0, 0) 0, V (0, 2) 0; On
V (0, 0) 0, V (0, 2) 0; On
2 3
2 2
x 0, 0 y 2 V (0, y) 8(0) y 4 0 y 0, no critical points,
2 2
y 0, 0 x 2 V ( x, 0) 8 x(0) 4 x 0 0, no critical points,
2
2 2 2 2 2
y 4 x , 0 x 2 V x, 4 x 8x 4 x 4 x 4 x 0
no critical points, V (0, 2) 0, V (2, 0) 0. Thus, there is a maximum volume of 64
3 3
2 2 2
3 3 3 .
if the box is
58. S( x, y, z) 2xy 2yz 2 xz;
xyz 27 z 27 S( x, y, z) 2xy 2y 27 2x 27 2 xy 54 54 ,
xy xy xy x y
x 0, y 0; S 54
x ( x , y ) 2 y 0 and S 54
2
y ( x , y ) 2 x
2
x
y
0 Critical point is (3, 3) z 3;
Sxx (3, 3) 4, S yy (3, 3) 4, Dxy (3, 3) 2 DxxDyy 2
D xy 12 0 and D xx 0 local minimum of
S(3, 3, 3) 54
59. Let x height of the box, y width, and z length,
cut out squares of length x from corner of the material
See diagram at right. Fold along the dashed lines to
form the box. From the diagram we see that the length
of the material is 2x y and the width is 2 x z . Thus
2 6 2x
xy
(2 x y)(2 x z) 12 z . Since
2x y
( , , ) ( , )
2xy 6 2x xy
V x y z x y z V x y
2x y
, where
x
0, y 0.
2
2 3 2 2 3
2
2 4 3 2 2
4 3y 4x y 4x y xy
2 12x 4x 4x y x y
Vx
( x, y ) 0 and V ( , ) 0
2
y x y critical points
2
(2 x y)
(2 x y)
are 3, 0 , 3, 0 , 1 , 4 and 1 , 4 . Only 3, 0 and 1 , 4 satisfy x 0 and y 0.
3 3
3 3
3 3
For 3, 0 : z 0;
2
Vxx 3, 0 0, Vyy 3, 0 2 3, Vxy 3, 0 4 3 VxxVyy Vxy
48 0
saddle point. For 1 , 4 : z 4 ; V 1 4 80 1 4 2 1 4 4
3 3 3
xx , , V
3 3 3 3
yy , , V ,
3 3 3 3
xy
3 3 3 3
2
V 16
xxVyy V xy 0 and V 0
3
xx local maximum of V 1 , 4 , 4 16
3 3 3 3 3
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