In linear programming, you write "max" or "min" before the objective function.  In nonlinear programming, however, one has to write "sup" (for supremum) or "inf" (for infimum).  This is because the supremum or infimum may not actually be attained.  Consider, for example:

inf x

subject to xy >= 1

x, y >= 0.

If we set y to some large positive real M, then we can set x to 1/M.  Then, we can make x as close to zero as we want, but we can't actually make x zero.  In other words, the infimum is zero, but it is not attainable.

This is one of the big differences between the linear and nonlinear cases.

Kamil Abdulrahman, do you mean the optimal behaviour of the function 1/x when x is close to (approaching) 0? For x=0 , we have no value of this function, and, of course, no derivative of any order. The behaviour can be described  by using the notion of the limit. The left-side limit of 1/x at x=0 is minus infinity, the right-side limit of 1/x at x=0 is plus infinity. We cannot even say that at x=0 there is a supremum equal to plus infinity or a infimum equal to minus infinity.

Thanks for you response Adam and Przemyslaw. I am having a deep thought on these class of function. Analytically, the concept of limit will only explain theoretical angle while the concept of infimum and supremum is also for like minds. In this era where most student are theoretically weak, an appropriate applicable areas will intrest student of Applied science and engineering.

Alternatively, can we look at this function from angle of fuzzy logic?

Kamil, so what is your point? To show to the students what practical meaning/application might be related with the function 1/x where x approaches zero?

Kamil, the students don't even need to know the terms "infimum" and "supremum" to understand the example that I gave.  You can just use "min" if you want.  You could also draw a picture of the feasible region (which is the region lying above the hyperbola y = 1/x), so that students can see what is happening visually.  The point is simply that, when nonlinear functions are involved, you have to be careful when trying to optimise.  All the students need is a familiarity with drawing a function of a single variable.

By the way, I'm afraid that I don't see any connection with fuzzy logic.  Fuzzy logic is concerned with performing inference when the membership of entities to discrete classes is uncertain.  It has nothing to do with either analysis or optimisation.