The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $R = \sqrt{3}$ is

Prove that the function $g(x) = \log x$ does not have any maxima or minima.

If $x=1$ and $x=2$ are extreme points of $f(x)=\alpha \log x+\beta x^2+x$,where $\alpha$ and $\beta$ are constants,then the value of $\alpha^2+2 \beta$ is

The height (in $cm$) of a cylinder of the greatest volume that can be inscribed in a sphere of radius $3 \ cm$ is

$A$ 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

The perimeter of a sector is $p$. What is its radius when the area of the sector is maximum?