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Math E21a – Fall 2009 – HW #8 problems
Read sections 11.8 and 12.1 (and maybe 12.212.3) and do the following problems:
Problems to turn in on Thurs, Oct 29
:
1. (Prob. 11.8/16) Use the Method of Lagrange Multipliers to find the maximum and minimum values of the
function
(, ,) 3
3
f xyz
x y
z
subject to the two constraints
0
xyz
and
22
21
xz
.
2. (Prob. 11.8/18) Find the extreme values of the function
(, ) 2
3
4 5
f xy
x
y
x
in the region described
by the inequality
16
xy
.
3. (Probs. 11.8/2122) a) The total production
P
of a certain product depends on the amount
L
of labor used and
the amount
K
of capital investment. In Sections 11.1 and 11.3 we discussed how the CobbDouglass model
1
Pb
LK
follows from certain economic assumptions, where b and
are positive constants and
1
.
If the cost of a unit of labor is
m
and the cost of a unit of capital is
n
, and the company can spend only
p
dollars as its total budget, then maximizing the production
P
is subject to the constraint
mL nK
p
.
Show that the maximum production occurs when
p
L
m
and
(1
)
p
K
n
.
b) If we now assume that the production is fixed at
1
bL K
Q
, where
Q
is a constant, what values of
L
and
K
will minimize the cost function
(, )
CLK
mL nK
?
4. (Prob. 11.8/24) Use the Method of Lagrange Multipliers to prove that the triangle with maximum area that
has a given perimeter
p
is equilateral. [
Hint
: Use Heron’s formula for the area:
()
A
ss x s y s z
,
where
2
p
s
and
x
,
y
, and
z
are the lengths of the sides of the triangle.]
5. (Prob. 11.8/38) Find the maximum and minimum volumes of a rectangular box whose surface area is
1500 cm
2
and whose total edge length is 200 cm. [
Note
: Due to the multiple constraints, it’s not hard to show
that in neither case will the box be a cube!]
6. (Prob. 11.8/40) The plane 4
3
8
5
intersects the cone
222
zxy
in an ellipse.
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 Fall '09
 
 Math

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