If $x + y = 10$,then the maximum value of $xy$ is

Find the maximum and the minimum values,if any,of the function given by $f(x) = x, x \in (0, 1)$.

The maximum volume of a right circular cylinder if the sum of its radius and height is $6 \text{ m}$ is: (in $\pi \text{ m}^3$)

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$ for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln(g(x))$ for all $x \in R$,then the number of points in $R$ at which $g$ has a local maximum is

The denominator of a fraction is $16$ more than the square of the numerator. Find the least value of the fraction.

The maximum value of $\left(\frac{1}{x}\right)^x$ is