For $a > 0$,if the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$ attains its maximum value at $p$ and minimum value at $q$ such that $p^2 = q$,then $a =$

What is the minimum value of the sum of a real number $x$ and its reciprocal?

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

It is given that at $x=1,$ the function $f(x) = x^{4}-62x^{2}+ax+9$ attains its maximum value on the interval $[0,2].$ Find the value of $a.$

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	$(S)$ $8$

$A$ rectangle with its sides parallel to the $X$-axis and $Y$-axis is inscribed in the region bounded by the curves $y=x^2-4$ and $y=\frac{4-x^2}{2}$. The maximum possible area of such a rectangle is closest to the integer.