Thomas Calculus 13th [Solutions]

Chapter 14 Practice Exercises 1067
2
fxx (0, 0) 12 x| (0,0) 0, f yy (0, 0) 12 y| (0,0) 0, fxy (0, 0) 3 fxx f yy f xy 9 0 saddle point width
f (0, 0) 0. For
maximum value of f 1 , 1 1
2 2 4
1 1
2
, : f 6, 6, 3 27 0
2 2 xx f yy fxy fxx f yy f xy and f xx 0 local
68.
69.
70.
2
fx ( x , y ) 3 x 3 y 0 and
2 2
f y ( x , y ) 3 y 3 x 0 y x and
4 3
x x 0 x ( x 1) 0 the critical
points are (0, 0) and (1, 1). For (0, 0): fxx (0, 0) 6 x| (0,0) 0, f yy (0, 0) 6 y| (0,0) 0, fxy
(0, 0) 3
2
fxx f yy f xy 9 0 saddle point with f (0, 0) 15. For (1, 1): fxx
(1, 1) 6, f yy (1, 1) 6,
2
fxx (1, 1) 3 fxx f yy f xy 27 0 and f xx 0 local minimum value of f (1, 1) 14
2
fx ( x , y ) 3 x 6 x 0 and
2
f y ( x , y ) 3 y 6 y 0 x ( x 2) 0 and y( y 2) 0 x 0 or x 2 and
y 0 or y 2 the critical points are (0, 0), (0, 2), ( 2, 0), and ( 2, 2). For (0, 0) :
2
fxx (0, 0) 6x 6| (0,0) 6, f yy (0, 0) 6y 6| (0,0) 6, fxy (0, 0) 0 fxx f yy f xy 36 0 saddle
point with f (0, 0) 0. For
2
(0, 2) : fxx (0, 2) 6, f yy (0, 2) 6, fxy (0, 2) 0 fxx f yy f xy 36 0 and
f xx 0 local minimum value of f (0, 2) 4. For ( 2, 0) : fxx
( 2, 0) 6, f yy ( 2, 0) 6
2
fxy ( 2, 0) 0 fxx f yy f xy 36 0 and f xx 0 local maximum value of f ( 2, 0) 4. For ( 2, 2) :
2
fxx ( 2, 2) 6, f yy ( 2, 2) 6, fxy ( 2, 2) 0 fxx f yy f xy 36 0 saddle point with f ( 2, 2) 0.
3 2
fx ( x, y) 4x 16x 0 4 x( x 4) 0 x 0, 2, 2; f y ( x, y) 6y 6 0 y 1. Therefore the
critical points are (0, 1), (2, 1), and ( 2,1). For (0, 1):
2
fxx
(0,1) 12x 16| (0,1) 16, f yy (0, 1) 6,
2
fxy (0,1) 0 fxx f yy f xy 96 0 saddle point with f (0, 1) 3. For (2, 1) : f xx (2, 1) 32,
2
f yy (2, 1) 6, fxy (2, 1) 0 fxx f yy f xy 192 0 and f xx 0 local minimum value of f (2, 1) 19.
For ( 2,1) :
2
fxx ( 2, 1) 32, f yy ( 2,1) 6, fxy ( 2, 1) 0 fxx f yy fxy
192 0 and f xx 0 local
minimum value of f ( 2,1) 19.
71. (i) On OA f ( x, y) f (0, y) 2
y 3y for 0 y 4
f (0, y) 2y 3 0 y 3 . But 0, 3
2
2
is not in the region.
Endpoints: f (0, 0) 0 and f (0, 4) 28.
(ii) On AB, f ( x, y) f ( x, x 4)
2
x 10x
28
for 0 x 4 f ( x, x 4) 2x
10 0
x 5, y 1. But (5, 1) is not in the region.
Endpoints: f (4, 0) 4 and f (0, 4) 28.
2
(iii) On OB, f ( x, y) f ( x, 0) x 3x for 0 x 4 f ( x, 0) 2x 3 x 3 and y 0 3 , 0
2
2
is a
critical point with f 3 , 0 9 . Endpoints: f (0, 0)
2 4
0 and f (4, 0) 4.
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