If $$f\left( x \right) = k{x^3} - 9{x^2} + 9x + 3$$       is monotonically increasing in every interval, then which one of the following is correct ?

Solution :
Given $$f\left( x \right) = k{x^3} - 9{x^2} + 9x + 3$$
On differentiating w.r.t. $$x$$, we get
$$f'\left( x \right) = 3k{x^2} - 18x + 9$$
For a function to be monotonically increasing.
$${b^2} - 4ac < 0$$
Here, $$a = 3k,\,\,b = - 18,\,\,c = 9$$
$$\eqalign{ & \therefore \,{b^2} - 4ac = {\left( { - 18} \right)^2} - 4\left( {3k} \right)\left( 9 \right) \cr & = \left( { - 18} \right)\left( { - 18} \right) - \left( {3k} \right)18 \times 2 \cr & \Rightarrow 36 - 12k < 0 \cr & \Rightarrow k > 3 \cr} $$