Question

For the cubic equation $x^3-3x^2+3x-1=0$, which of the following is/are true

• A. All $3$ roots are equal
• B. Only one root is repeated
• C. It is increasing in $x\in R$
• D. It is decreasing in $x\in R$

Answer 1

Answer: (A), (C)

It is increasing in $x\in R$

$f\left(x\right)=x^3-3x^2+3x-1=(x-1{)}^3$
$\Rightarrow f^{\prime }\left(x\right)=3(x-1{)}^2$
Clearly $f^{\prime }\left(x\right)\ge 0\forall x\in R$
$\Rightarrow$ It is increasing $\forall x\in R$
Also $f^{\prime }\left(x\right)=0$ has equal roots
$\Rightarrow$All roots of the given cubic equation are equal.