Applications Of Derivatives Question 167
Question: The values of -a- for which the function $ (a+2)x^{3}-3ax^{2}+9ax-1 $ decreases monotonically throughout for all real x, are
[Kurukshetra CEE 2002]
Options:
A) $ a<-2 $
B) $ a>-2 $
C) $ -3<a<0 $
D) $ -\infty <a\le -3 $
Show Answer
Answer:
Correct Answer: D
Solution:
If $ f(x)=(a+2)x^{3}-3ax^{2}+9ax-1 $ decreases monotonically for all $ x\in R, $ then $ f’(x)\le 0 $ for all $ x\in R $
therefore $ 3(a+2)x^{2}-6ax+9a\le 0 $ for all $ x\in R $
therefore $ (a+2)x^{2}-2ax+3a\le 0 $ for all $ x\in R $
therefore $ a+2<0 $ and Discriminant $ \le 0 $
therefore $ a<-2 $ , $ -8a^{2}-24a\le 0 $
therefore $ a<-2 $ and $ a(a+3)\ge 0 $
therefore $ a<-2 $ , $ a\le -3 $ or $ a\ge 0 $
therefore $ a\le -3 $
therefore $ -\infty <a\le -3 $ .
BETA
