HW10
–
gilbert
–
(55035)
1
This print-out should have 16 questions.
Multiple-choice 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
P
Q
0
R
2
1
0
-1
-2
-3
A. local maximum at Q,
B. local minimum at R,
C.
local minimum at P .
1.
A and C only
2.
A and B only
3.
B and C only
4. C only correct
5. none of them
6. A only
7. B only
8. all of them
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.
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 .
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
+ 3xy
−
2 .
1. local max at (1,
−
1),
local min at (0, 0)
2. local max at (1,
−
1),
saddle point at (0, 0)
3. local min at (0, 0),
saddle point at (1,
−
1)
4. local min at (1,
−
1),
saddle point at (0, 0) correct
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 .
But f
X
= 0 when
∂
f
∂
x
=
3x
2
+ 3y
=
0
,
i.e.,
y
=
−
x
2
,

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