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

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(- π / 2, 0)

(0, π)

(- π / 2, π / 2)

none of these

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detailed solution

Correct option is B

Consider the function $f (x)$ given by

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

We observe that

$f (0) = f (π) = 0$

Therefore, 0 and $π$ 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 $(0, π) .$

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