A function $$f\left( x \right) = 1 - {x^2} + {x^3}\,\,$$ is defined in the closed interval $$\left[ { - 1,1} \right].$$ The value of $$x,$$ in the open interval $$(-1,1)$$ for which the mean value theorem is satisfied, is

The maximum area (in square units) of a rectangle whose vertices lie on the ellipse $${x^2} + 4{y^2} = 1\,\,$$ is

Which one of the following graphs describes the function? $$f\left( x \right) = {e^{ - x}}\left( {{x^2} + x + 1} \right)\,?$$

A vector $$\overrightarrow P $$ is given by $$\,\,\overrightarrow P = {x^3}y\overrightarrow a {}_x - {x^2}{y^2}\overrightarrow a {}_y - {x^2}yz\overrightarrow a {}_z.\,\,\,$$ Which one of the following statements is TRUE?