If $x + y = 8$,find the maximum value of $xy$.

$A$ poster is to be printed on a rectangular sheet of paper of area $18 \ m^2$. The margins at the top and bottom of $75 \ cm$ each and at the sides $50 \ cm$ each are to be left. Then the dimensions,i.e.,height and breadth of the sheet,so that the space available for printing is maximum,are respectively:

The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function $f(x) = x^3 + 3x - 9$ on the interval $[-2, 3]$. If the difference between the first and the second term of the progression is equal to $f'(0)$,then the common ratio of the $G.P.$ is

Let $\exp(x)$ denote the exponential function $e^x$. If $f(x) = \exp\left(x^{\frac{1}{x}}\right)$ for $x > 0$, then the minimum value of $f$ in the interval $[2, 5]$ is:

If the function $f(x) = \frac{x}{5} + \frac{5}{x}, (x \neq 0)$ attains its relative maximum value at $x = a$, then $\sqrt{a^2 + 2a - 6} = $

The local maximum value of the function $f(x)=-(x-2)^3(x+2)^2$ is