Question

The number of roots of the equation $x^3+x^2+2x+\sin x=0$ in $(-2\pi ,2\pi )$ are

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

The correct option is D 1

Let f(x)$=x^3+x^2+2x+sinx$.
Now f'(x)$=3x^2+2x+2+cosx$
$=3(x+\frac{1}{3}{)}^2+\frac{5}{3}+cosx>0$ for all $x$
Which implies that $f\left(x\right)$ is a monotonically increasing function.
$f\left(0\right)=0$. Hence, there can be no other root of $f\left(x\right)=0$.
Hence, option D is correct