utility – Intuition for proper monotne functions

While reading documents about utility theory, I came across a definition of a suitable monotonous function, which is a function with $ u & # 39;> 0 $, $ u & # 39; & # 39; <0 $, $ u & # 39; & # 39; & # 39;> 0 $ and so.
The first conditions are clear: more is better than less and marginal utility decreases. I am not sure how to interpret the higher order conditions and, in general, what is "appropriate" about this function?

Any intuition is appreciated.