A random variable $X$ has a zero probability of taking non-positive values but has a non-zero probability of taking values in any range $[0, x]$ for any $x> 0$. The curve describing the probability density function forms an arc of a circle. Which of these are possible shapes (ignoring the scale) for the probability density function $f(x)$? Identify clearly the mathematical reasons, using the correct terminology, for your answers.
If the radius of the circle forming the arc of the pdf is $1$, what is the maximum value that the random variable could possibly take?