A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is $\tan^{-1}\left(\frac{2}{3}\right)$. Water is poured into it at a constant rate 6 cubic metre per hour. The rate at which the level of the water is rising at the instant when the depth of water in the tank is 5 meter is

Water is running into a conical vessel, 18cm deep and 6cm in radius, at the rate of 0.1 $cm^3$/sec. If the water is 4cm deep, then the water level rising at the rate

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

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

The point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate, is

Let height and radius of a right circular cylinder are $h$ and $r$ respectively. If it is open at the top, having a given surface area and greatest volume, then

If x is real, then minimum value of $x^2 - 8x + 17$ is :