question_answer

The value of a for which the function\[(a+2){{x}^{3}}-\]\[-3a{{x}^{2}}+9ax-1\] decrease monotonically throughout for all real x, are

A)  \[a<-2\]                              

B)  \[a>-2\]

C)  \[-3<a<0\]                         

D)  \[-\infty <a<-3\]

Correct Answer: D

Solution :

 For the function \[f(x)=(a+2){{x}^{3}}-3a{{x}^{2}}+9ax-1\] \[f'(x)=3(a+2){{x}^{2}}-6ax+9a\] \[\because \]Function is monotonically decreasing \[\forall x\in R\] \[\therefore \]\[f'(x)\le 0\,\forall x\in R\] \[3(a+2){{x}^{2}}-6ax+9a\le 0\,\forall x\in R\] \[(a+2){{x}^{2}}-2ax+3a\le 0\,\forall x\in R\] This is only possible when coefficient of x2 is less than zero and discriminate is negative. \[\Rightarrow \]\[a+2<0\]and\[4{{a}^{2}}-4(a+2)(3a)<0\] \[\Rightarrow \]\[a<-2\]and\[4{{a}^{2}}-12{{a}^{2}}-24a<0\] \[\Rightarrow \]\[a<-2\]and\[-8{{a}^{2}}-24\,a<0\] \[\Rightarrow \]\[a<-2\]and\[-8a(a+3)<0\] \[\Rightarrow \]\[a<-2\]and\[a(a+3)>0\] \[\therefore \]Required solution is \[(-\infty ,-3).\]