The concepts of extrema and critical points of functions is an extremely useful concept in mathematics, particularly the areas of calculus and differential equations. Extrema may be absolute, such as the maximum (or highest) value or the minimum (or lowest) value attained by the function over its defined domain, or extrema may be local, such as the maximum or minimum value attained by a function in a particular, localized interval of the domain. Critical points are any interior points of the domain at which the derivative of the function is either zero or undefined (that is, does not exist). Importantly, on all closed intervals (that is, finite intervals that include boundary points), continuous functions will take both an absolute maximum and an absolute minimum; it is impossible to construct a continuous function over a closed interval that does not take on absolute...