rhodes (ajr2283) – HW14 – Gilbert – (56380)
1
This
printout
should
have
9
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
001
10.0 points
In
the
contour
map
below identify
the
points
P, Q
, and
R
as local minima, local
maxima, or neither.
3
2
1
0
1
2
0
3
2
1
0
1
2
Q
P
R
A.
local maximum at
Q
,
B.
local minimum at
R
,
C.
local maximum at
P
.
1.
all of them
2.
B only
3.
A only
4.
C only
5.
none of them
correct
6.
A and B only
7.
B and C only
8.
A and C only
Explanation:
A.
FALSE: the point
Q
lies on the 0
contour and this contour divides the region
near
Q
into two regions.
In one region the
contours have values increasing to 0, while in
the other the contours have values decreasing
to 0.
So the surface does not have a local
minimum at
Q
.
B.
FALSE: the contours near
R
are closed
curves enclosing
R
and the contours
increase
in value as we approch
R
. So the surface has
a local maximum at
R
, not a local minimum.
C.
FALSE: the contours near
P
are closed
curves enclosing
P
and the contours
decrease
in value as we approch
P
. So the surface has
a local minimum at
P
, not a local maximum.
keywords:
contour
map,
local
extrema,
True/False,
002
10.0 points
Locate and classify all the local extrema of
f
(
x, y
) =
x
3
−
y
3
+ 3
xy
−
1
.
1.
local min at (1
,
−
1),
saddle point at (0
,
0)
correct
2.
local max at (1
,
−
1),
local min at (0
,
0)
3.
local max at (1
,
−
1),
saddle point at (0
,
0)
4.
local max at (0
,
0),
saddle point at (1
,
−
1)
5.
local min at (0
,
0),
saddle point at (1
,
−
1)
Explanation:
Since
f
has derivatives everywhere, the crit
ical points occur at the solutions of
∇
f
(
x, y
) =
f
x
i
+
f
y
j
= 0
.
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 Spring '07
 TextbookAnswers
 Critical Point, Multivariable Calculus, Fermat's theorem, local minimum

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