Question

Consider the equation $a{z}^2+z+1=0$ having purely imaginary root where $a=\cos \theta +i\sin \theta ,i=\sqrt{-1}$ and function $f\left(x\right)=x^3-3x^2+3(1+\cos \theta )x+5,$ then answer the following questions. Which of the following is true about $f\left(x\right)$?

• A. $f\left(x\right)$ decreases for $x\in [2n\pi ,(2n+1\left)\pi \right],n\in Z$
• B. $f\left(x\right)$ decreases for $x\in \left[\right(2n-1\left)\frac{\pi }{2},(2n+1)\frac{\pi }{2}\right],n\in z$
• C. $f\left(x\right)$ is non-monotonic function
• D. $f\left(x\right)$ increases for $x\in R$.

Answer 1

The correct option is D $f\left(x\right)$ increases for $x\in R$.

$a{z}^2+z+1=0$ ...(1)
Where $a=\cos \theta +i\sin \theta$
let $z_1\&z_2$ be the roots of the eq. (1)
Since equation (1) have purely imaginary root.
$\therefore z_1=-\overline{z_1}\&z_2=-\overline{z_2}$
sum of the roots of the eq. (1)=$z_1+z_2=\frac{1}{a}\Rightarrow \overline{z_1}+\overline{z_2}=\frac{1}{\overline{a}}$
$\frac{1}{a}+\frac{1}{\overline{a}}=0$ ...{ $\because z_1=-\overline{z_1}\&z_2=-\overline{z_2}$}
$\Rightarrow \cos \theta =0$ ...{ $\because a=\cos \theta +i\sin \theta$}
$\Rightarrow \cos \theta =0$ ...(2)

$f\left(x\right)=x^3{-3x}^2+3(1+\cos \theta )x+5$

$\Rightarrow f\left(x\right)=x^3{-3x}^2+3x+5$ ...{ from 1}
$\Rightarrow f\left(x\right)={(x-1)}^3+4$

$\because f^{\prime }\left(x\right)=3{(x-1)}^2>0$
Therefore f(x) is increasing for $x\in R$
Hence, option 'D' is correct.