The minimum value of $f(a) = (2a^2 - 3) + 3(3 - a) + 4$ is

$A$ manufacturer can sell $x$ items at a price of rupees $\left(5 - \frac{x}{100}\right)$ each. The cost price of $x$ items is $\text{Rs} \left(\frac{x}{5} + 500\right)$. Find the number of items he should sell to earn maximum profit.

For the function $f(x) = x^4(12\ln x - 7)$,match the following columns:

Column-$I$	Column-$II$
$(A)$ If $(a, b)$ is a point of inflection,then $a - b$ equals	$(P)$ $3$
$(B)$ If $e^t$ is the point of local minimum,then $12t$ equals	$(Q)$ $1$
$(C)$ If the graph is concave down on $(d, e)$,then $d + 3e$ equals	$(R)$ $4$
$(D)$ If the graph is concave up on $(p, \infty)$,then $p$ equals	$(S)$ $8$

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$,then

The function $f(x) = \sin x (1 + \cos x)$ is maximum at $x = \dots$

The ratio of the height of a cone of maximum volume inscribed in a sphere to its radius is