Thomas Calculus 13th [Solutions]

(iii) On CD,
3
f ( x, y) f (1, y) y 3y 2 for 1 y 1
Endpoints: f (1,1) 6 and f (1, 1) 2.
(iv) On AD,
3
f ( x, y) f ( x, 1) x 3x for 1 x 1
Chapter 14 Practice Exercises 1071
2
f (1, y) 3y 3 0 no solution.
2
f ( x, 1) 3x 3 0 x 1 and y 1
yielding the corner points ( 1, 1) and (1, 1) with f ( 1, 1) 2 and f (1, 1) 2
(v) For the interior of the square,
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
x x 0 x 0 or x 1 y 0 or y 1 (0, 0) is an interior critical point of the square region
with f (0, 0) 1;( 1, 1) is on the boundary. Therefore the absolute maximum is 6 at (1, 1) and the
absolute minimum is 2 at (1, 1) and ( 1, 1).
79.
2
f 3x i 2yj and g 2xi
2y
2y 2y 1 or y 0.
CASE 1:
CASE 2:
Evaluations give
j so that
2 2
f g 3x i 2 yj (2xi 2 yj ) 3x 2x
and
2
1 3x 2x x 0 or x 2
3 ; x 0 y 1 yielding the points (0, 1) and (0, 1);
2
5
3 3
x y yielding the points
2 5
3 3
, and
2 5
3 3
, .
2
y 0 x 1 0 x 1 yielding the points (1, 0) and ( 1, 0).
2 5 23
3 3 27
f (0, 1) 1, f , , f (1, 0) 1, and f ( 1, 0) 1. Therefore the absolute
maximum is 1 at (0, 1) and (1, 0), and the absolute minimum is 1 at ( 1, 0).
80. f yi xj and g 2xi 2yj so that f g yi xj (2xi 2 yj ) y 2 x and xy 2 y
2
x 2 (2 x) 4 x x 0 or
2
4 1.
CASE 1: x 0 y 0 but (0, 0) does not lie on the circle, so no solution.
2
CASE 2: 4 1
1
or
1
2
2 . For 1 2 2 2
, y x 1 x y 2x x y 1 yielding the
2 2
points
1
,
1
and
1
,
1
. For
2 2
2 2
y x yielding the points
1
,
1
and
1
,
1
.
2 2 2 2
1 2 2 2
, y x 1 x y 2x x 1
and
2 2
Evaluations give the absolute maximum value f 1
,
1 f 1
,
1 1
and the absolute minimum
value f 1 1 f 1 1 1
2 2 2 2 2
, , .
2 2 2 2
2
81. (i)
2 2
2 2
f ( x, y) x 3y 2y on x y 1 f 2 xi (6y
2) j and g 2xi
2yj so that
f g 2 xi (6y 2) j (2xi 2 yj ) 2x 2x
and 6y 2 2y 1 or x 0.
CASE 1: 1 6y 2 2y y 1
and
CASE 2:
2
3
x yielding the points
2
2
x 0 y 1 y 1 yielding the points (0, 1).
3 1 1
2 2 2
3
,
1
.
2 2
Evaluations give f , , f (0, 1) 5, and f (0, 1) 1. Therefore 1 and 5 are the extreme
2
values on the boundary of the disk.
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2014 Pearson Education, Inc.