Sometimes we would like to to get a rough sketch of a function around some point $x=a$, but the function is very "flat" at that point - after we work out the first few derivatives we get f(a) = f'(a) =f''(a) = f^{(3)}(a) = f^{ (4)}(a) = 0 , which means that, assuming finally $ f^{(5)} \neq 0 $, the function behaves like $ \frac{ f^{(5)}(a)}{5!} (x-a)^5 $
near that point. Hence if the $5$-th derivative is positive at $a$, the shape is approximately like that of $x^5$ at the origin, and if the derivative is negative then the shape is like that of $-x^5$ at the origin.
One example I can think of when this arises is in the study of the stability of equilibrium points of differential equations. The stablity of the equilibria is determined by the behaviour near $x=a$, which is partly determined by the sign of the first non-zero derivative evaluated at the equilibrium point.