If $f(x) = 3x + \frac{12}{x}$ is continuous on $R - \{0\}$ and $M$ is its local maximum value,then $\lim_{x \rightarrow M} f(x) = $

The radius of the cylinder of maximum volume,which can be inscribed in a sphere of radius $R$ is :-

The maximum slope of the curve $y=-x^{3}+3x^{2}+2x-27$ is

Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x-2) + \alpha x^2 + 4x - \alpha, x > 2$, 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 $M =$

If $x=-2$ and $x=4$ are the extreme points of $y=x^3-\alpha x^2-\beta x+5$,then