Question

If the function f(x) = kx³ − 9x² + 9x + 3 is monotonically increasing in every interval, then
(a) k < 3
(b) k ≤ 3
(c) k > 3
(d) k ≥ 3

Answer 1

(c) k > 3

$$\begin{gathered} f\left(x\right)=k{x}^3-9x^2+9x+3 \\ f\text{'}\left(x\right)=3k{x}^2-18x+9 \\ =3\left(k{x}^2-6x+3\right) \\ \text{Given:}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) is monotonically increasing in every interval.} \\ \Rightarrow f\text{'}\left(x\right)>0 \\ \Rightarrow 3\left(k{x}^2-6x+3\right)>0 \\ \Rightarrow \left(k{x}^2-6x+3\right)>0 \\ \Rightarrow k>0\mathrm{and}{\left(-6\right)}^2-4\left(k\right)\left(3\right)<0\left[\because a{x}^2+bx+c>0\Rightarrow a>0\mathrm{and}\mathrm{Disc}<0\right] \\ \Rightarrow k>0\mathrm{and}{\left(-6\right)}^2-4\left(k\right)\left(3\right)<0 \\ \Rightarrow k>0\mathrm{and}36-12k<0 \\ \Rightarrow k>0\mathrm{and}12k>36 \\ \Rightarrow k>0\mathrm{and}k>3 \\ \Rightarrow k>3 \end{gathered}$$