$20 \ m$ of wire is available to fence a flowerbed in the form of a circular sector. If the flowerbed is to have maximum surface area,then the radius of the circle is (in $m$)

If $x + 2y = 8$,then the maximum value of $xy$ is .......

Let $f : R \rightarrow R$ be a function defined by $f(x) = ||x+2|-2|x||$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$,then $m+n$ is

Find the local maximum and local minimum values for the function given by $f(x) = x^{2}$.

The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi-vertical angle $\alpha$ is one-third that of the cone and the greatest volume of the cylinder is $\frac{4}{27} \pi h^{3} \tan^{2} \alpha$.