1
WINTER 2004 QUIZ TH RE E MATH 200
Problem: Find the absolute maximum and absolute minimum values of
f(x, y) = 3xy −6x−3y+ 7
on the closed rectangular region Rwith vertices (0,0),(3,0),(3,5), and (0,5).
Note:
compare with the example in the text, page 997. Here is a plot of the function fover the rectangle. It is clear
for the plot that the extrema occur at the two corners.
0
0.5
1
1.5
2
2.5
3
x
012345
y
–10
–5
0
5
10
15
How to Find the Absolute Extrema of a Continuous Function fof Two Variables on a Closed and Bounded Set
R
1. Find the critical points of fthat lie in the interior of R.
2. Find all boundary points at which the absolute extrema can occur.
3. Evaluate the function f(x, y)at the points obtained in steps (1) and (2). The largest of these values is the
absolute maximum and the smallest is the absolute minimum.
Answer: We £rst £nd the critical points in the interior of the rectangle. To do this, we need to £nd the partial derivatives
of f(x, y)and set them to zero. Note that
∂f
∂x = 3y−6 = 0,∂f
∂y = 3x−3 = 0.
Hence, there is only one critical point Pcthat occurs at (1,2). The value of f(1,2) is 6−6−6 + 7 = 1.
We must check the extreme values on each line segment of the boundary of R.
•Case 1:L1: 0 ≤x≤3and y= 0. On L1,fbecomes f(x, 0) = −6x+ 7 which has a unique minimum at
(3,0) with value −11 and unique maximum at (0,0) with value 7.
•Case 2:L2: 0 ≤y≤5and x= 3. On L2,fbecomes f(3, y) = 9y−18 −3y+ 7 = 6y−11 which has a
unique minimum at (3,0) with value −11 and unique maximum at (3,5) with value 19.
•Case 3:L3: 0 ≤x≤3and y= 5. On L3,fbecomes f(x, 5) = 15x−6x−15 + 7 = 9x−8which as a
unique minimum at (0,5) with value −8and unique maximum at (3,5) with value 19. Note: the value at (3,5)
is consistent with Case 2.
•Case 4:L4: 0 ≤y≤5and x= 0. On L4,fbecomes f(0, y) = −3y+ 7 which has a unique minimum at
(0,5) with value −8and unique maximum at (0,0) with value 7.
