Why maximize the reciprocal instead of minimizing the negative? Is there some property which makes it preferable to find the maximum $1/x$ of x in a set, rather than directly finding the minimum of x in a set? 
The context is a homework project, where the maximizing the reciprocal is hinted at in the problem description. We have several factors to weigh in making a decision for an AI agent. 
I realize this is a very general question but that's all there is to it. Does anyone have any ideas?
Update: Maybe what I mean to ask is: If each x is a function of several parameters, is there some reason to use reciprocals in combining the parameters?
 A: There's not a general rule that says it's better to do one than the other.  But sometimes in particular cases it's better to do one than the other.
For example, to find the value of $x$ that minimizes $\sqrt{x^2 + (x-1)^2}$, it's simpler to find the value that minimizes $x^2 + (x-1)^2$, since the derivative of $\sqrt{x^2 + (x-1)^2}$ is more work to compute (and in this case, you can find the value of $x$ that minimizes the quadratic polynomial just by completing the square.
As another example, in Regiomontanus' angle maximization problem, you want to find the maximum angle, but since the tangent function is increasing whenever the angle is between $0$ and a right angle, you can maximize the tangent of the angle instead.  And in that particular problem, the derivative of the tangent of the angle is simpler than the derivative of the angle itself.
As yet another example, in statistics, maximizing the logarithm of a likelihood function is often a lot less work than maximizing the likelihood function directly.
A: Maximizing $1/x$ is not quite the same as minimizing $x$.  Let $S$ be a set of real numbers.  Finding $\max\{1/x \mid x \in S\}$ is equivalent to finding  the smallest positive $x \in S$ (if such exists), while $\min\{x \mid x \in S\}$ is simply the smallest $x$ in $S$ (if such exists), which may very well be negative.
