Lecture Note Function of Two Variables
)
In this case each of the second partials is a constant and will have that constant value
at(6, 3. Thus,
()() ()()226, 3 6, 3 6, 3
22 1 41 3 0
Df=−xx fyy ⎣⎡fxy ⎤⎦=×−− =−=>SinceD> 0 , we check the sign offxx(6, 3)to determine whether (6, 3)yields a local
minimum or a local maximum. And sincefxx(6, 2)= 2 > 0 , the critical point ()
yields a local minimum value. This value is
6, 3
f(6, 3)= 622 −× 6 3 3+ 9 6+= −× 5 − 22Example 4
A company produces and sells two styles of umbrellas. One style sells for $20 each
and the other sells for $25 each. The company has determined that if x thousand of the
first style and y thousand of the second style are produced, then the total cost in
thousands of dollars is given by the function
()^223
,33 32297
2
Cxy=−+ +−+x xy y x y 0How many of each style of umbrella should the company produce and sell in order to
maximize profit?
Solution
Since x thousand umbrellas sell for $20 each and y thousand umbrellas sell for $25
each, the revenue function (in thousands of dollars) is given byRxy( ,2025)=+x y
Thus the profit function is
() ()( )()(^223)
2
(^223)
2
,,,
20 25 3 3 32 29 70
3 3 12 54 70
Pxy Rxy Cxyxyxxyy xyxxyy x y=−
=+− −++−+
=− + − − + −
The first partial derivatives are
Pxyx=− + − (^6312) and Pxyy= 3354 −+
Now solve the following system of equations.
63120
14, 32
33540
xy
xy
xy
⎧−+ −=
⎨ ⇒= =
⎩ −+=
The company will make the maximum profit if it produces and sells 14,000 of the first
style of umbrella and 32,000 of the second style. (The student can verify (with the D-
Test) that the profit is indeed a maximum at (14, 32).)
Example 5
The only grocery store in a small rural community carries two brands of frozen orange
juice, a local brand that it obtains at the cost of 30 cents per can and a well-known
national brand that it obtains at the cost of 40 cents per can. The grocer estimates that
if the local brand is sold for x cents per can and the national brand for y cents per can,
approximately 70 − 5 x+ 4 ycans of the local brand and 80 + 6 x− 7 ycans of the
national brand will be sold each day. How should the grocer price each brand to
maximize the profit from the sale of the juice? (Assume that the absolute maximum
and the relative maximum of the profit function are the same.)
(Answer: xy=53, = 55 )