Two particles move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity $u$ and the second starts from rest with constant acceleration $f$. Then,

The points on the curve $y = 12x - x^3$ at which the gradient is zero are

$A$ 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

Find the local maximum and local minimum values for the function given by $g(x) = \frac{x}{2} + \frac{2}{x}, \quad x > 0$.

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

If the function $f$ given by $f(x) = x^3 - 3(a - 2)x^2 + 3ax + 7$ for some $a \in R$ is increasing in $(0, 1]$ and decreasing in $[1, 5)$,then a root of the equation $\frac{f(x) - 14}{(x - 1)^2} = 0$ $(x \neq 1)$ is