Question

Find the value of $a$ for which $f\left(x\right)=a{x}^3-9a{x}^2+9x+3$ is increasing on $R$.

Answer 1

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

$f^{\prime }\left(x\right)>0$ for $f\left(x\right)$ to be increasing

$\therefore f^{\prime }\left(x\right)=3a{x}^2-18ax+9>0$

$\therefore a{x}^2-6ax+3>0$

For a quadratic equation to be greater than $0$,

$a>0$ & $D<0$

$\therefore (6a{)}^2-4(a\left)\right(3)<0$

$36a^2-12a<0$

$3a^2-a<0$

$a(3a-1)<0$

$a\in (0,\frac{1}{3})$