Answer:
(a) f(x) = x2
+ 2x ? 5
f(x) = 2x +
2 = 2 (x + 1)
f?(x) = 0 x
= ?1
Intervals are and (?1, )
when then f?(x)
< 0
when then f?(x)
> 0
is strictly
increasing in (?1, )
and strictly decreasing in.
f(x)
= 10 ? 6x ? 2x2
Intervals
are
when
then
f?(x) = ?2 (?) > 0
when
then
f?(x) = ?2 (+) < 0
is
strictly increasing in and
strictly decreasing in
(a) f(x) = ?2x3
? 9x2 ? 12x + 1
f?(x) = ?6x2
? 18x ? 12
= ?6 (x2 +
3x + 2)
= ?6 (x + 1) (x + 2)
f?(x) = 0
Intervals are :
Where
When
f?(x) = (?6) (?)
(+)> 0
When
f?(x) = (?6) (+)
(+) < 0
is strictly
increasing in (?2, ?1) and strictly decreasing in .
(b) f(x) = 6 ? 9x ? x2
f?(x) = ?9
?2x = ?(2x + 9)
f?(x) = 0
Intervals are :
When
f?(x) = (?)
(?) > 0
When
f?(x) = (?) (+)
< 0
is
strictly increasing in and
strictly decreasing in .
(c) g(x) = (x + 1)3
(x ? 3)3
f?(x) = (x
+ 1)3.3(x ? 3)2 + (x ? 3)3 3 (x + 1)2
= 3 (x + 1)2
(x ? 3)2 [x + 1 + x ?3]
= 6 (x + 1)2
(x ? 3)2 (x ? 1)
f?(x) = 0 x = ?1, 1,
3.
Intervals are : and (3, )
When
f?(x)
= 6(+) (+) (?) < 0
When ,
f?(x)
= 6(+)(+) (?) < 0
When
When ,
f?(x)
= 5 (+) (+) (+) > 0
is
strictly increasing in (1, 3) (3,
) and
strictly decreasing in
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