Question

Find the value of a for which the function $f\left(x\right)=a{x}^3-3(a+2)x^2+9(a+2)x-1$ is decreasing for all $x\in R$.

Answer 1

Given, $f\left(x\right)=a{x}^3-3(a+2)x^2+a(a+2)x-1$

A function is decreasing $\forall x\in R$ if $f^{\prime }\left(x\right)<0\forall x\in R$

$f^{\prime }\left(x\right)=3a{x}^2-6(a+2)x+9(a+2)$

$3a{x}^2-6(a+2)x+9(a+2)<0$

$\Rightarrow a{x}^2-2(a+2)x+3(a+2)<0\forall x\in R$

This is possible only where its discriminant

is less than zero and coefficient of $x^2$ is less

than zero.

$\Rightarrow △<0$ & $a<0$

$\Rightarrow 4(a+2{)}^2-4a.3(a+2)<0$ & $a<0$

$\Rightarrow a^2+4a+4-3a^2-6a<0$ & $a<0$

$\Rightarrow -2a^2-2a+4<0$ & $a<0$

$\Rightarrow a^2+a-2>0$ & $a<0$

$\Rightarrow a^2+2a-a-2>0$ & $a<0$

$\Rightarrow (a+2)(a-1)>0$ & $a<0$

$\Rightarrow a<-2$ or $a>1$ & $a<0$.

$\Rightarrow a<-2$.

$\therefore a<-2$ makes $f\left(x\right)$ decreasing function

$\forall x\in R$