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The equation $\sin x+x \cos x=0$ has atleast one root in the interval

Option: 1
$\left(-\frac{\pi}{2}, 0\right)$

Option: 2
$(0, \pi)$

Option: 3
$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Option: 4
None of these

Answers (1)

best_answer

Consider the function $f(x)$ is given by

$f(x)=\int(\sin x+\cos x) d x=x \sin x$

We observe that

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

Therefore, $0\: \text{and}\: \pi$ are two roots of $f(x)=0$ .

Consequently, $f(x)=0$ i.e., $\sin x+x \cos x=0$ has at least one root in $(0,\pi)$ .

Posted by

Kshitij

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