Find the point on the curve $$\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8$$
that is farthest from the point $$(0, -2)$$.

Let $$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$$ Find the intervals in which $$\lambda $$ should lie in order that $$f(x)$$ has exactly one minimum and exactly one maximum.

Find all the tangents to the curve
$$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel to the line $$x+2y=0$$.

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