If $f(x)=3x^3-9x^2-27x+15$,then the maximum value of $f(x)$ is $.....$

Let a function $f(x) = \begin{cases} -\ln(3x - [3x]) & ; 3x \neq n, n \in N \\ \ln(\operatorname{sgn}(3x)) & ; 3x = n, n \in N \end{cases}$,where $[.]$ and $\operatorname{sgn}(x)$ denote the greatest integer function and signum function respectively. Then the number of points where $f(x)$ is minimum in $x \in (0, 5)$ is:

The greatest value of the real-valued function $f(x)=(x+1)^{1/3}-(x-1)^{1/3}$ on the interval $[0, 1]$ is:

The cost of running a bus from $A$ to $B$ is $Rs. \left( av + \frac{b}{v} \right)$,where $v \text{ km/h}$ is the average speed of the bus. When the bus travels at $30 \text{ km/h}$,the cost is $Rs. 75$,while at $40 \text{ km/h}$,it is $Rs. 65$. The most economical speed (in $\text{km/h}$) of the bus is:

Statement-$I$: Among the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6}, 7^{1/7}$,the maximum is $3^{1/3}$.
Statement-$II$: The function $f(x) = x^{1/x}$ increases for $0 < x < e$ and decreases for $x > e$.

$A$ wire of length $20 \ m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in $meters$) of the hexagon,so that the combined area of the square and the hexagon is minimum,is: