Math 2224 Review Extrema
For each of the functions
†
f(x,y)
and regions
†
R
, answer the questions below.
a. Find the critical point(s) of
†
f(x,y)
.
b. Use the Second Partials Test to determine whether there is a local maximum, local minimum,
or saddle point at each critical point.
c. Find the absolute maximum and absolute minimum for each function over the region
†
R
.
1.
†
f(x,y)=xy +x, R
is the region bounded by the triangle whose vertices are
†
0, 0
( )
, 1, 1
( )
, 2, 0
( )
.
2.
†
f(x,y)=x3-3x2+3y2+y3, R
is the region bounded by
†
y=x, y= - x, and x=3.
3.
†
f(x,y)=1
x+xy +1
y, R
is the region bounded by
†
y=1
x, y=1, x=1, and x=2.
Answers:
1. a. (0, -1) b. saddle point c. absolute maximum =
†
9
4
; absolute minimum = 0.
2. a. (0, 0), (2, 0), (0, -2), (2, -2) b. (0, 0) saddle point; (2, 0) local minimum; (0, -2) local
maximum; (2, -2) saddle point; c. absolute maximum = 54; absolute minimum = -4.
3. a. (1,1) b. local minimum c. absolute maximum = 3.5; absolute minimum = 3