If a right circular cone having maximum volume is inscribed in a sphere of radius $3 \, cm$,then the curved surface area (in $cm^2$) of this cone is

$20$ is divided into two parts such that the product of the cube of one part and the square of the other part is maximum. The parts are:

$A$ square piece of tin of side $30\,cm$ is to be made into a box without a top by cutting a square of side $x$ from each corner and folding up the flaps to form a box. If the volume of the box is maximum,then its surface area (in $cm^2$) is equal to $............$.

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum,then $x^2+y^2=$

The displacement $S$ of a particle measured from a fixed point $O$ on a line is given by $S = t^3 - 16t^2 + 64t - 16$. Then the time at which displacement of the particle is maximum is

Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12$,$x \in (-4,4)$. If $M$ is the local maximum value of the function $f$ in $(-4,4)$,then $M =$