The maximum value of $${{\ln x} \over x}$$ is

A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is

What is the length of the longest interval in which the function $$f(x) = 3\sin x - 4{\sin ^3}x$$ is increasing?

Let $$f(x) = x + {1 \over x}$$, where x$$\in$$ (0, 1). Then which one of the following is correct?