If $x$ is real,then the greatest and least values of $\frac{x^2 - x + 1}{x^2 + x + 1}$ are

The radius (in $cm$) of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone with radius $10 \ cm$ is

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

If $x$ and $y$ are two positive real numbers such that $xy = 4$, then the minimum value of $\left(\sqrt{x} + \frac{y^2}{2}\right)$ is

If the function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$,where $a > 0$,attains its local maximum and local minimum values at $p$ and $q$ respectively,such that $p^2=q$,then $f(3)$ is equal to:

$A$ cylindrical container is to be made from a certain solid material with the following constraints: It has a fixed inner volume of $V \ mm^3$,has a $2 \ mm$ thick solid wall,and is open at the top. The bottom of the container is a solid circular disc of thickness $2 \ mm$ and has a radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is $10 \ mm$,then the value of $\frac{V}{250 \pi}$ is