Question

The number of roots of the equation $x^3+x^2+2x+\sin x=0in[-2\pi ,2\pi ]$ is

• A. $1$
• B. $2$
• C. $3$
• D. None of these

Answer 1

Answer: (A)

$1$

Let $f\left(x\right)=x^3+x^2+2x+sinx$.
$f\left(0\right)=0$
On differentiating the function we get,
$f^{\prime }\left(x\right)=3x^2+2x+2+cosx$
The minimum value of $3x^2+2x+2$ is $\frac{15}{9}$ . Hence, $f^{\prime }\left(x\right)$ is always positive. The function is monotonically increasing.
Hence, $x=0$ is the only root of $f\left(x\right)=0$. Hence, it has one real root in $[-2\pi ,2\pi ]$