If the function f : is defined by f ( x ) = | x | ( x - sin x ) , then which of the following statements is TRUE ? [2020]

Question

If the function $f : ℝ \to ℝ$ is defined by $f (x) =$ $| x |$ $(x - \sin x),$ then which of the following statements is TRUE ? [2020]

Smart Achievers · Accepted Answer

(3)

$f (x) is a non-periodic, continuous and odd function$

$f (x) = {\begin{matrix} - x^{2} + x \sin x, & x < 0 \\ x^{2} - x \sin x, & x \geq 0 \end{matrix}$

$f' (x) =$ ${\begin{matrix} - 2 x + \sin x + x \cos x, & x < 0 \\ 2 x - \sin x - x \cos x, & x \geq 0 \end{matrix}$

$f'' (x) =$ ${\begin{matrix} - (x - \sin x) - x (1 - \cos x), & x < 0 \\ (x - \sin x) + x (1 - \cos x), & x > 0 \end{matrix}$

$∵$ $x - \sin x < 0$ $if x < 0$ and

$1 - \cos x > 0,$ $\forall x \in ℝ$

$∴$ $- (x - \sin x) - x (1 - \cos x) > 0$ $if x < 0$

and $(x - \sin x) + x (1 - \cos x) > 0$ $if x > 0$

$\Rightarrow f' (x) > 0,$ $\forall x \in ℝ$ $\Rightarrow f (x) is increasing in ℝ$

$\Rightarrow f (x) is one-one$

$∴$ $\lim_{x \to - \infty} (- x^{2}) (1 - \frac{\sin x}{x}) = - \infty$

$∴$ $\lim_{x \to \infty} x^{2} (1 - \frac{\sin x}{x}) = \infty$

$\Rightarrow Range of f (x) = ℝ$

$\Rightarrow f (x) is an onto function.$

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