The sufficient conditions for the function $f:R \to R$ to have a local maximum at $x = a$ 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

$A$ closed cylinder of given volume will have the least surface area when the ratio of its height to its base radius is:

If $A = \{x \in R : \frac{\pi}{4} \leq x \leq \frac{\pi}{3}\}$ and $f(x) = \sin x - x$,then $f(A)$ is equal to

The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of diameter $6$ units is

If local maximum of $f(x) = \frac{ax + b}{(x - 1)(x - 4)}$ exists at $(2, -1)$, then $a + b =$