Question

The function$f\left(x\right)={\left[x\right(x-2\left)\right]}^2$is increasing in the set

• A. $(-\infty ,0)\cup (2,\infty )$
• B. $(-\infty ,1)$
• C. $(0,1)\cup (2,\infty )$
• D. $(1,2)$
• E. $(0,2)$

Answer 1

The correct option is C

$(0,1)\cup (2,\infty )$

Explanation for correct option

$f\left(x\right)={\left[x\right(x-2\left)\right]}^2$

$f\text{'}\left(x\right)=4x(x-1)(x-2)$

for $f\left(x\right)$ is increasing $f\text{'}\left(x\right)>0$

$$\begin{gathered} \Rightarrow 4x(x-1)(x-2)>0 \\ \therefore x\in (0,1)\cup (2,\infty ) \end{gathered}$$

Hence, option C is correct.