moseley (cmm3869) – HW14 – Gilbert – (56380)
1
This printout should have 9 questions.
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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
P
,
C.
local minimum at
R
.
1.
C only
2.
A and C only
3.
A and B only
4.
all oF them
5.
A only
6.
B and C only
7.
B only
correct
8.
none oF them
Explanation:
A.
±ALSE: 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. TRUE: 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
.
C. ±ALSE: 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.
keywords:
contour map, local extrema,
True/±alse,
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 (0
,
0),
saddle point at (1
,
−
1)
2.
local max at (1
,
−
1),
local min at (0
,
0)
3.
local min at (1
,
−
1),
saddle point at (0
,
0)
correct
4.
local max at (1
,
−
1),
saddle point at (0
,
0)
5.
local max 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
 Sadler
 Critical Point, Multivariable Calculus, Fermat's theorem, local minimum

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