Find all maxima and minima of the function $$$y = x{\left( {x - 1} \right)^2},0 \le x \le 2$$$
Also determine the area bounded by the curve $$y = x{\left( {x - 1} \right)^2}$$,
the $$y$$-axis and the line $$y-2$$.

Investigate for maxima and minimum the function $$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$$

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.