The number of points in (- \infty , \infty), for which x2 – x sin x

Engineering

Mathematics

Monotonicity

Question

The number of points in ( $- \infty, \infty$ ), for which x² – x sin x – cos x = 0, is

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f (x) = x² – x sin x – cos x

f (x) is an even function

and f (0) = – 1

and f ' (x) = x(2 – cos x) > 0 for x $\in$ (0, $\infty$ )

$∴$ f (x) is an increasing function

$∴$ f (x) must have exactly one real root in (0, $\infty$ )

$\Rightarrow$ two real roots.

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