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 $a$

Let $f\left(x\right)=x^3–3a{x}^2+(27a^2+9)x+2016f^{\prime }\left(x\right)=3x^2-6ax+27a^2+9\Rightarrow f^{\prime }\left(x\right)=3(x^2-2ax+a^2)-3a^2+27a^2+9\Rightarrow f^{\prime }\left(x\right)=3(x-a{)}^2+24a^2+9$

As $f^{\prime }\left(x\right)>0\forall x\in R$
So $f\left(x\right)$ is monotonic increasing function.
Hence exactly one real root for any real $a$