What is the local maximum value of $\frac{\log x}{x}$?

The point $(0, 5)$ is closest to the curve ${x^2} = 2y$ at

$A$ right triangle is drawn in a semicircle of radius $R = \frac{1}{2}$ with one of its legs along the diameter. The maximum area of the triangle is

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:

All the points on the curve $y^{2}=4a|x+a \sin(x/a)|$,where the tangent is parallel to the $x$-axis,lie on:

Let $\sqrt{3}$ be the radius and $\frac{\pi}{3}$ be the semi-vertical angle of a given cone. Then the height of the right circular cylinder of maximum volume that can be inscribed in the given cone is