Find all solutions of $$4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$$

Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$
where the tangent to the curve has the greatest slope.

Show that $$1+x$$ $$In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $$ for all $$x \ge 0$$

The function $$y = 2{x^2} - In\,\left| x \right|$$ is monotonically increasing for values of $$x\left( {x \ne 0} \right)$$ satisfying the inequalities ......... and monotonically decreasing for values of $$x$$ satisfying the inequalities ............