If the height of a cone of greatest volume that can be inscribed in a sphere of radius $R$ is $kR$,then the ratio of the volume of the cone to the volume of the sphere is

The function $f(x) = x^{25}(1 - x)^{75}$ for $x \in [0, 1]$ attains its maximum at $x = \dots$

The maximum value of the function $f(x)=3x^3-18x^2+27x-40$ on the set $S=\{x \in R : x^2+30 \leq 11x\}$ is

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$ for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln(g(x))$ for all $x \in R$,then the number of points in $R$ at which $g$ has a local maximum is

Show that of all the rectangles inscribed in a given fixed circle,the square has the maximum area.

Let $M$ and $m$ be respectively the local maximum and the local minimum values of the function $f(x) = 2x^3 - 9x^2 + 12x + 5$ in the interval $[0, 3]$. Then $M - m$ is equal to