Question

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

Answer 1

$f\left(x\right)>0$ for all $x\in R$

$3k{x}^2-18kx+9>0$ for all $x\in R$

$k{x}^2-6kx+3>0$ for all $x\in R$

$k>0$ and $36k^2-12k>0$

$k>0$ and $12k(3k-1)<0$

$k>0$ and $k(3k-1)<0$

$3k-1<0$

$k<\frac{1}{3}$

$k\in (0,1/3)$