Question

The set of all real values of $a$ for which the function $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$ decreases monotonically throughout for all real $x,$ is

• A. $a<-2$
• B. $a>-2$
• C. $-3\le a<0$
• D. $a\le -3$

Answer 1

The correct option is D $a\le -3$

We have $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$
$\Rightarrow f^{\prime }\left(x\right)=3(a+2)x^2-6ax+9a=3\left[\right(a+2)x^2-2ax+3a]$
Now, $f^{\prime }\left(x\right)\le 0\forall x\in R$
$\Rightarrow a<-2$ and discriminant $\le 0$
$\Rightarrow 4a^2\le 12a(a+2)$
$\Rightarrow a^2\le 3a^2+6a$
$\Rightarrow 2a^2+6a\ge 0$
$\Rightarrow 2a(a+3)\ge 0$
$\Rightarrow a\in (-\infty ,-3]\cup [0,\infty )$
But $a<-2$. Hence $a\in (-\infty ,-3]$