7.2 Root-finding (optimization)

In many situations, we want to find the solution to $f(x)=0$. Many problems of optimization (such as finding maxima and minima) can be cast in this form. (Finding the $x$ that maximizes [or minimizes] $f(x)$ is equivalant to finding $x$ such that $g(x)=f'(x)=0$.) The independent variable $x$ may actually be a vector representing a large number of independent variables but we'll start with one-dimensional examples.