If the function $f\left(x\right) = K{x}^3 – 9x^2 + 6x + 3$ is increasing $\forall x\in R then K\in$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{kx}}^3-9{\mathrm{x}}^2+6\mathrm{x}+3 \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=3{\mathrm{kx}}^2-18\mathrm{x}+6 \\ \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{increasing} \end{gathered}$$

$$\begin{gathered} \Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)>0 \\ 3{\mathrm{kx}}^2-18\mathrm{x}+6>0 \\ {\mathrm{kx}}^2-6\mathrm{x}+2>0 \\ ∆<0\Rightarrow 36-4\left(\mathrm{k}\right)\left(2\right)<0 \end{gathered}$$

$$\begin{gathered} 36-8\mathrm{k}<0 \\ \mathrm{k}>\frac{36}{8} \\ \mathrm{k}>\frac{9}{2} \\ \mathrm{k}\in \left(\frac{9}{2}, \infty \right) \end{gathered}$$