for $x > 0$ , which of the following is true?

sin x < x

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sin x \leq x

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s i n x \geq x

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sin x > x

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Solution

The correct option is B $sin x < x$
Let $y = f (x) = x - sin x$
$\Rightarrow \frac{d y}{d x} = 1 - cos x = 2 {sin}^{2} \frac{x}{2} > 0$
Thus $f (x)$ is increasing function for $x > 0$
$\Rightarrow f (x) > f (0)$
$\Rightarrow x - sin x > 0$
$\Rightarrow x > sin x$

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