Question

The values of $k$ for which the function $f\left(x\right)=k{x}^3-9x^2+9x+3$ may be increasing on $R$ are

• A. $k>3$
• B. $k<3$
• C. $k\le 3$
• D. None of these

Answer 1

Answer: (A)

$k>3$

We have, $f\left(x\right)={kx}^3-9x^2+9x+3$

$\Rightarrow f^{\prime }\left(x\right)=3{kx}^2-18x+9$

Since $f\left(x\right)$ is increasing on $R,$ therefore,

$f^{\prime }\left(x\right)>0\forall x\in R$

$\Rightarrow 3{kx}^2-18x+9>0\forall x\in R$

$\Rightarrow {kx}^2-6x+3>0\forall x\in R$

$\Rightarrow k>036-12k<0$

$[\because {ax}^2+bx+c>0\forall x\in R\Rightarrow a>0$ and $D<0]$

$\Rightarrow k>3$

Hence, $f\left(x\right)$ is increasing on $R,$ if $k>3.$