Question

The polynomial equation $x^3-3a{x}^2+(27a^2+9)x+2016=0$ has

• A. Exactly one real root for any real $a$
• B. Three real roots for any real $a$
• C. Three real roots for any $a\ge 0$, and exactly one real root for any $a<0$
• D. Three real roots for any $a\le 0$, and exactly one real root for any $a>0$

Answer 1

The correct option is A Exactly one real root for any real $a$

$f\left(x\right)=x^3-3a{x}^2+(27a^2+9)x+2016=0$

$f^{\prime }\left(x\right)=3x^2-\left(6a\right)x+(27a^2+9)$

$D=(-69{)}^2-4(3\left)\right(27a^2+9)$

$=27a^2-327a^2-108$

$=-(300a^2+108)$ $\to$ Value is always -ve for all '$a$'

$\Rightarrow$ If $f\left(x\right)<0$ for $x\in R,a\in R$

$f\left(x\right)B$ strictly decreasing

its graph will somewhat be line this

$\Rightarrow$ only one real root for any real '$a$'