The number of points in , for which x 2 - x sin x - cos x = 0 , is [2013]

Question

The number of points in $(- \infty, \infty),$ for which $x^{2} - x \sin x - \cos x = 0,$ is [2013]

Smart Achievers · Accepted Answer

(3)

Let $f (x) = x^{2} - x \sin x - \cos x$

$∴$ $f' (x) = 2 x - x \cos x = x (2 - \cos x)$

$∴$ $f is increasing on (0, \infty) and decreasing on (- \infty, 0)$

$Also \lim_{x \to \infty} f (x) = \infty, \lim_{x \to - \infty} f (x) = \infty, and f (0) = - 1$

$∴$ $y = f (x) meets x-axis twice.$

$i.e., f (x) = 0 has two points in (- \infty, \infty) .$

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