MA 22S1
Assignment 7
Due 7-9 December 2015
Id:
22S1-f2015-7.m4,v 1.2 2016/01/09 19:32:26 john Exp john
1. Find the tangent plane to the surface
x2 + 2xy + 3xz + 4y 2 + 5yz + 6z 2 + 7x + 8y + 9z = 164
at (1, 2, 3).
Solution:
∇f (x, y, z) = (2x + 2y + 3z + 7, 2x + 8y + 5z + 8, 3x + 5y + 12z + 9)
and
∇f (1, 2, 3) = (22, 41, 58),
so the tangent plane is
(22, 41, 58) · (x − 1, y − 2, z − 3) = 0
or
22x + 41y + 58z = 278.
2. (a) Minimise x2 + 2xy + 3xz + 4y 2 + 5yz + 6z 2 + 7x + 8y + 9z subject
to x + 2y + 3z ≤ c.
Note: You can assume the existence of such a minimum to be
given.
Solution: With
f (x, y, z) = x2 + 2xy + 3xz + 4y 2 + 5yz + 6z 2 + 7x + 8y + 9z
and
g(x, y, z) = x + 2y + 3z
1
Id:
22S1-f2015-7.m4,v 1.2 2016/01/09 19:32:26 john Exp john 2
we have
∇f (x, y, z) = (2x + 2y + 3z + 7, 2x + 8y + 5z + 8, 3x + 5y + 12z + 9),
∇g(x, y, z) = (1, 2, 3).
Then ∇f = λ∇g is equivalent to the system
2x+2y+3z+7 = λ,
2x+8y+5z+8 = 2λ,
3x+5y+12z+9 = 3λ.
The solution to this system is
11λ − 299
,
82
x=
y=
9λ − 21
,
82
z=
14λ + 22
.
82
Substituting,
71λ − 275
71λ2 − 2063
, g(x, y, z) =
.
164
82
There are two possibilities. Either λ = 0 and g(x, y, z) ≤ c or
λ ≤ 0 and g(x, y, z) ≤ c. In the first case the minimum is at
f (x, y, z) =
x=−
299
,
82
y=−
21
,
82
z=
22
,
82
λ = 0,
2063
275
, g(x, y, z) = −
.
164
82
≤ c. In the second case the minimum
f (x, y, z) = −
This only works if − 275
82
occurs where
so
71λ − 275
=c
82
82c + 275
71
The choice of the sign is dictated by the condition λ ≤ 0. Then
λ=
11c − 222
11λ − 299
=
,
82
71
9λ − 21
9c + 12
y=
=
,
82
71
14c + 66
14λ + 22
=
,
z=
82
71
82c + 275
λ=
,
71
41c2 + 275c − 432
71λ2 − 2063
=
,
f (x, y, z) =
164
71
g(x, y, z) = c.
x=
This only works if c ≤ − 275
.
82
Id:
22S1-f2015-7.m4,v 1.2 2016/01/09 19:32:26 john Exp john 3
(b) Check that the derivative of the minimum value is equal to the
Lagrange multiplier, as claimed in lecture.
Solution: The minimum value is, as shown in the previous part,
min
f (x, y, z) =
g(x,y,z)≤c
(
− 2063
162
3362c2 +22550c−35424
5751
if − 275
≤ c,
82
if c ≤ − 275
.
82
Differentiating gives
d
min f (x, y, z) =
dc g(x,y,z)≤c
This is just λ, as expected.
(
0
82c+275
71
≤ c,
if − 275
82
if c ≤ − 275
.
82