At this point, it is important to note that the above derivations can be applied with little extra work to multi-dimensional optimization problems since they do not explicitly require 1D assumptions. Thus, they are inherently different from 1D search techniques such as bisection, Brent's method, etc. In the following chapter, we apply the above techniques to a specific multi-dimensional problem, conditional density estimation which is of particular interest. However, it is not the only place where the above work could be useful and many other possibilities remain to be investigated (some are now closer at hand than others).