If $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{kx}}^3-9{\mathrm{x}}^2+9\mathrm{x}+3$ is monotonically increasing in R, then

$\begin{array}{c}Given f\left(x\right)=k{x}^3-9x^2+9x+3\\ diff w.r.to x on both sides\\ f^{\text{'}}\left(x\right)=3k{x}^2-18x+9\\ \sin ce f\left(x\right) is monotonically increa\sin g then f^{\text{'}}\left(x\right)\ge 0\\ \\ \Rightarrow 3k{x}^2-18x+9\ge 0\end{array}$

For a Q.E to be always greater than or equal to  zero then roots of Q.E must be complex.

$\begin{array}{c}\Rightarrow {\mathrm{b}}^2-4\mathrm{ac}\le 0\\ \Rightarrow 324-108k\le 0\\ \Rightarrow 108k\ge 324\\ \Rightarrow \mathrm{k}\ge 3\end{array}$