question_answer 22)

Find the intervals in which the following functions are strictly increasing or strictly decreasing : (a) x2 + 2x – 5 (b) 10 – 6x – 2x2 (c) –2x3 – 9x2 – 12x +1 (d) 6 – 9x – x2 (e) (x +1)3 (x – 3)3  

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