Question

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

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

Answer 1

The correct option is C

$k>3$

Explanation for correct option:

Find the value of $k$:

Given expression,

$f\left(x\right)=k{x}^3–9x^2+9x+3$

Now

$$\begin{gathered} f\text{'}\left(x\right)=3k{x}^2–18x+9 \\ =3[k{x}^2–6x+3] \end{gathered}$$

Since, the function is monotonically increasing, then $f\text{'}\left(x\right)>0$.

For this, discriminant $D<0$ and $k>0$

$$\begin{gathered} D<0 \\ \Rightarrow b^2–4ac<0, \\ \Rightarrow 36–12k<0 \\ \Rightarrow k>3 \end{gathered}$$

Hence, the correct option is C.