Question

The function $f\left(x\right)=2x^3-9x^2+12x-6$ is monotonic decreasing, when

• A. $1<x<2$
• B. $x>2$
• C. $x<1$
• D. None of these

Answer 1

The correct option is A

$1<x<2$

Explanation for correct option

Given function $f\left(x\right)=2x^3-9x^2+12x-6$

$\Rightarrow f\text{'}\left(x\right)=6x^2-18x+12$

We know that the condition for $f\left(x\right)$ to be monotonically decreasing is $f\text{'}\left(x\right)<0$.

$$\begin{gathered} \Rightarrow f\text{'}\left(x\right)<0 \\ \Rightarrow 6x^2-18x+12<0 \\ \Rightarrow x^2-3x+2<0 \\ \Rightarrow x^2-2x-x+2<0 \\ \Rightarrow x(x-2)-1(x-2)<0 \\ \Rightarrow (x-2)(x-1)<0 \\ \Rightarrow x\in \left(1,2\right) \end{gathered}$$

Thus, $f\left(x\right)$ decreases in the interval $(1,2)$.

Hence, the correct option is $\left(\mathrm{A}\right)$ .