JEE Advanced 2020 Paper 1Question 1
If the function $f:\mathbb{R}\to\mathbb{R}$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?
Functions
Solution
$f$ is odd. For $x>0$, $f(x)=x^2-x\sin x$, $f\'(x)=2x-\sin x-x\cos x=x(1-\cos x)+(x-\sin x)>0$, so $f$ is strictly increasing on $\mathbb R$ (one-one), and $f\to\pm\infty$ as $x\to\pm\infty$, so it is onto.