Mark the correct alternative in the following:


Mark the correct alternative in the following:

If the function $f(x)=k x^{3}-9 x^{2}+9 x+3$ is monotonically increasing in every interval, then

A. $k<3$

B. $k \leq 3$

C. $k>3$

D. $k<3$


Formula:- (i) $a x^{2}+b x+c>0$ for all $x \Rightarrow a>0$ and $b^{2}-4 a c<0$

(ii) $a x^{2}+b x+c<0$ for $a l l x \Rightarrow a<0$ and $b^{2}-4 a c<0$

(iii) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$


$f(x)=k x^{3}-9 x^{2}+9 x+3$

$\frac{d(f(x))}{d x}=f^{\prime}(x)=3 k x^{2}-18 x+9$

for increasing function $f^{\prime}(x)>0$


$\Rightarrow 3 k x^{2}-18 x+9>0$

$\Rightarrow k x^{2}-6 x+3>0$

using formula (i)

$36-12 k<0$

$\Rightarrow k>3$

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