1. The answer is (e). The curves meet in three places, (

2
,
4)
,
(0
,
0) and (3
,
9). The
region between the curves is made up of a pair of leafshaped regions touching
at the point (0
,
0). The
x
intervals for the two regions are [

2
,
0] and [0
,
3]. In
the ﬁrst region,
y
goes from
x
2
to
x
3

6
x
, while in the second region it is the
other way around. Therefore, (c) would be right except that I forgot to change
the
x
limits on the second region; the correct answer is therefore (e).
2. The answer is (b). In (a) there is nothing distinguishing a maximum from a
minimum. In (c) it is possible that the minimum is elsewhere on the boundary,
for example at the origin. In (d) there is not a minimum in the interior, but it
could be on one of the other boundary segments. To see that it is (b) and not
(e), notice that if
f
x
and
f
y
are both greater than zero, then there can be no
maximum on the interior (because the gradient does not vanish), on the interior
of the
x
axis segment (because
f
x
>
0), on the interior of the
y
axis (because
f
y
>
0), or at the origin (because
D
u
>
0 for any
u
pointing into the triangle).
Any continuous function on a closed bounded set has a maximum (see page
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 Fall '07
 Temkin
 Critical Point, Multivariable Calculus, Gradient

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