CM221A
ANALYSIS I
Solutions to Exercise Sheet 5
Calculate the derivatives of each of the following two functions.
Find all of the
local and global maximum and minimum values, if there are any, of the functions
on the intervals given.
Sketch the graphs of the functions in these regions
and
explain your results!
1.
f
(
x
) =
1
1+log(
x
)
2
on (0
,
e];
f
(
x
)
is positive and vanishes as
x
→
+
∞
and also as
x
→
0 + 0
. It is not
defined at
0
and so has a g.l.b. but not a min. Its only turning point is where
f
0
(
x
) = 0
, i.e. at
x
= 1
, which is its maximum. It has a local min at
x
= e
,
which is an end point of the interval.
2.
f
(
x
) =
xe

x
2
/
2
on [0
,
∞
);
It is positive everywhere on the interval except at
x
= 0
, where it vanishes.
Thus it has a local and global minimum at
x
= 0
.
It converges to zero as
x
→
+
∞
. It has a maximum where its derivative vanishes, at
x
= 1
.
3. Evaluate the following derivative (you must state the conditions under which
the result holds)
d
d
x
f
(
x
)
2
g
(
x
)
3
.
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 Spring '09
 Calculus, Derivative, Continuous function, 4 g, 0 2 G, 2 3g, g.l.b.

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