A) \[a<-2\]
B) \[a>-2\]
C) \[-3<a<0\]
D) \[-\infty <a\le -3\]
Correct Answer: D
Solution :
If \[f(x)=(a+2){{x}^{3}}-3a{{x}^{2}}+9ax-1\] decreases monotonically for all \[x\in R,\]then \[f'(x)\le 0\]for all \[x\in R\] Þ \[3(a+2){{x}^{2}}-6ax+9a\le 0\]for all \[x\in R\] Þ \[(a+2){{x}^{2}}-2ax+3a\le 0\]for all \[x\in R\] Þ \[a+2<0\]and Discriminant\[\le 0\] Þ \[a<-2\],\[-8{{a}^{2}}-24a\le 0\] Þ \[a<-2\]and\[a(a+3)\ge 0\] Þ \[a<-2\], \[a\le -3\]or \[a\ge 0\]Þ \[a\le -3\]Þ\[-\infty <a\le -3\] .
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