Calculus- Early Transcendentals, 2021a

593
13.6.5 f x = 3cos(3x)cos(2y), f y = −2sin(3x)sin(2y), 13.7.13 The sides and bottom should all be 2/3 meter, and
f xy = −6cos(3x)sin(2y), f yy = −4sin(3x)cos(2y),
the sides should be bent up at angle π/3.
f xx = −9sin(3x)cos(2y)
13.7.14 (3,4/3)
13.6.6 f x = e x+y2 , f y = 2ye x+y2 , f xx = e x+y2 ,
f yy = 4y 2 e x+y2 + 2e x+y2 , f xy = 2ye x+y2
13.7.16 |b| if b ≤ 1/2, otherwise √ b − 1/4
3x 2
13.7.17 |b| if b ≤ 1/2, otherwise √ b − 1/4
13.6.7 f x =
2(x 3 + y 4 ) , f y = 2y3
x 3 + y 4 ,
f xx =
3x
x 3 + y 4 − 9x 4
2(x 3 + y 4 ) 2 , f yy = 6y2
x 3 + y 4 − 8y 6 13.7.19 1024/ √ 3
(x 3 + y 4 ) 2 ,
f xy = −6x2 y 3
13.8.1 a cube, √ 3
1/2 × 3√ 1/2 × 3√ 1/2
(x 3 + y 4 ) 2
13.8.2 65/3 · 65/3 · 130/3 = 2 · 65 3 /27
13.6.8 z x = −x
16z , z y = −y
4z , z xx = − 16z2 + x 2
16 2 z 3 ,
13.8.3 It has a square base, and is one and one half times as
z yy = − 4z2 + y 2
16z 3 , z xy = −xy
tall √ as wide. √ If the volume is V the dimensions are
3 2V/3 ×
3
64z 3
2V/3 × 3√ 9V/4.
13.6.9 z x = − y + z
x + y , z y = − x + z
x + y , z xx = 2
y + z
13.8.4 |ax 0 + by 0 + cz 0 − d|/ √ a 2 + b 2 + c 2
(x + y) 2 ,
z yy = 2
x + z
(x + y) 2 , z xy = 2z
13.8.5 (0,0,1), (0,0,−1)
(x + y) 2
13.8.6 3√ 4V × 3√ 4V × 3√ V/16
13.7.1 minimum at (1,−1)
13.8.7 Farthest: (− √ 2, √ 2,2 + 2 √ 2); closest: (2,0,0),
13.7.2 none
(0,−2,0)
13.8.8 x = y = z = 16
13.7.3 none
13.8.9 (1,2,2)
13.7.4 maximum at (1,−1/6)
13.8.10 ( √ 5,0,0), (− √ 5,0,0)
13.7.5 none
13.8.11 standard $65, deluxe $75
13.7.6 minimum at (2,−1)
13.8.12 x = 9, φ = π/3
13.7.7 f (2,2)=−2, f (2,0)=4
13.8.13 35, −35
13.7.8 a cube 1/ 3√ 2onaside
13.8.14 maximum e 4 , no minimum
13.7.9 65/3 × 65/3 × 130/3
13.8.15 5, −9/2
13.7.10 It has a square base, and is one and one half times as
13.8.16 3, 3, 3
tall √ as wide. √ If the volume is V the dimensions are
3 2V/3 ×
3
2V/3 × 3√ 9V/4.
13.8.17 a cube of side length 2/ √ 3
13.7.11 √ 100/3
14.1.1 16
13.7.12 |ax 0 + by 0 + cz 0 − d|/ √ a 2 + b 2 + c 2 14.1.2 4