If $x+y=6, x \geqslant 0, y \geqslant 0$,then the maximum value of $x^2 y$ is

Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is $\sin ^{-1}\left(\frac{1}{3}\right)$.

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is .....

$x$ and $y$ are two variables such that $x > 0$ and $xy = 1$. Then the minimum value of $x + y$ is

$A$ wire of length $28 \, m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

The global maximum value of $f(x) = \log_{10}(4x^3 - 12x^2 + 11x - 3)$,$x \in [2, 3]$,is