If $20$ is divided into two parts such that the product of the cube of one part and the square of the other part is maximum,then these two parts are:

$A$ rectangular sheet of tin $45 \, cm$ by $24 \, cm$ is to be made into a box without a top by cutting off a square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum (in $, cm$)?

$A$ wire $40 \text{ m}$ in length is to be cut into two pieces. One piece is formed into a square and the other piece into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum?

The minimum value of $e^{(2x^2 - 2x + 1)\sin^2 x}$ is

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is

If $f(x)$ is a function such that $f^{\prime}(x)=(x-1)^{2}(4-x),$ then