Question

The solution of the inequality $x^3(x-1{)}^2<0$ will be

• A. $x\in (1,\infty )$
• B. $x\in (0,1)$
• C. $x\in \varphi$
• D. $x\in (-\infty ,0)$

Answer 1

The correct option is D $x\in (-\infty ,0)$

Given the inequality as: $x^3(x-1{)}^2<0$

Let $f\left(x\right)=x^3(x-1{)}^2$
Now, the pivot points are $x=0\&1$
On the number line, we can draw them as:
$f\left(2\right)=2^3(2-1{)}^2=8>0$
$\therefore x^3(x-1{)}^2>0\forall x\in \text{region}(i)$

$(x-1{)}^2$ has an even power
$\therefore$ Same sign for region $\left(ii\right)$

$x^3$ has an odd power
$\therefore$ Sign changes for region $\left(iii\right)$

Now, we can plot the wavy curve as:

$\therefore x^3(x-1{)}^2<0\Rightarrow x\in (-\infty ,0)$