For $x + y = 8$,what is the maximum value of $xy$?

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:

$A$ triangular garden is fenced on two sides,and the third side is a straight river bank. The two fenced sides have an equal length of $x$. What is the maximum area that can be enclosed by the garden?

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$, then the ordered pair $(m, M)$ is equal to

Find the local maximum and local minimum values for the function $f(x) = x\sqrt{1 - x}$ where $0 < x < 1$.

$A$ square piece of tin of side $30\,cm$ is to be made into a box without a top by cutting a square of side $x$ from each corner and folding up the flaps to form a box. If the volume of the box is maximum,then its surface area (in $cm^2$) is equal to $............$.