The maximum value of $xy$ when $x+2y=8$ is

Divide $10$ into two parts such that the sum of double of the first and the square of the second is minimum.

The maximum volume of a right circular cylinder if the sum of its radius and height is $6 \text{ m}$ is: (in $\pi \text{ m}^3$)

If $x, y$ are two positive integers such that $x+y=20$ and the maximum value of $x^3 y$ is $k$ at $x=\alpha, y=\beta$,then $\frac{k}{\alpha^2 \beta^2} =$

The greatest value of the real-valued function $f(x)=(x+1)^{1/3}-(x-1)^{1/3}$ on the interval $[0, 1]$ is:

Let $V_1$ be the volume of a given right circular cone with $O$ as the centre of the base and $A$ as its apex. Let $V_2$ be the maximum volume of the right circular cone inscribed in the given cone whose apex is $O$ and whose base is parallel to the base of the given cone. Then,the ratio $V_2 / V_1$ is