The equation sin⁡x+xcos⁡x=0   has at least one root in the interval

# The equation $\mathrm{sin}x+x\mathrm{cos}x=0$   has at least one root in the interval

1. A

$\left(-\pi /2,0\right)$

2. B

$\left(0,\pi \right)$

3. C

$\left(-\pi /2,\pi /2\right)$

4. D

none of these

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### Solution:

Consider the function  given by

$f\left(x\right)=\int \left(\mathrm{sin}x+x\mathrm{cos}x\right)dx=x\mathrm{sin}x$

We observe that

$f\left(0\right)=f\left(\pi \right)=0$

Therefore, 0 and $\pi$ are two roots of $f\left(x\right)=0$

Consequently   has at least one

root in

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