Here's a plot of a function:
Nothing special about this function - it's just $y = \sin{x}$. It's immediately clear, even without doing any calculations, which parts of the curve has positive first derivative, and which parts has positive second derivative. Parts of the curve that are increasing have positive first derivative, so e.g. between $-1$ and $+1$ the first derivative must be positive. Similarly if the function is concave up or down, it must have negative and positive second derivative respectively. So we can immediately say that between $0$ and $3$ the second derivative is negative.
How can I quickly tell where the 3rd and 4th derivatives are positive or negative? In this case it's easy to just do the math, but I'm looking to interpret what a result for "the fourth derivative is positive" or "the fourth derivative is more positive at $x = 3$ than at $x = 4$" says about the shape of the underlying function. Strictly, I am only interested in the fourth derivative, but I'm guessing that insight with the third derivative will help illustrate what happens with the fourth derivative.