what is the lowest upper bound of standard deviation

kakarukeys

for a normalized distribution function f(p) over a finite interval I,

f(p) >= 0,
Int f(p) dp (over I) = 1

the standard deviation is given by the square root of

<p^2> - <p>^2

= Int p^2 f(p) dp - (Int p f(p) dp)^2

how to find the lowest upper bound (if exists) of the standard deviation?
Variational Calculus?