Consider a square of side $2 \ cm$. It is cut from one of its corners as shown in the adjacent figure. The maximum value of the sum of the perimeters of the two plane figures thus formed is

The least value of the sum of any positive real number and its reciprocal is

The global maximum value of $f(x) = \log_{10}(4x^3 - 12x^2 + 11x - 3)$,$x \in [2, 3]$,is

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $(x \ln x) f'(x) + (\ln x + 1) f(x) \geq 1$ for all $x \in [2,4]$,with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:
$(A): f(x) \leq 1$ for all $x \in [2,4]$
$(B): f(x) \geq \frac{1}{8}$ for all $x \in [2,4]$
Then,

The sum of two numbers is $20$. If the product of the square of one number and the cube of the other is maximum,then the numbers are:

$A$ closed vessel tapers to a point both at its top $E$ and its bottom $F$ and is fixed with $EF$ vertical. When the depth of the liquid in it is $x \, \text{cm}$,the volume of the liquid in it is $V(x) = x^2 (15 - x) \, \text{cu. cm}$. The length $EF$ is ........ $\text{cm}$.