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comment on compact spaces and maxima, with a hopefully useful example. please remove if this section isn't helpful.
The [[minima]] of a function can be calculated in an analogous manner.
The [[minima]] of a function can be calculated in an analogous manner.
The computation of maxima of functions remains an important issue in more advanced mathematics. For example, it is often useful to consider real-valued continuous functions on more exotic topological spaces, and ask what their maxima are. Note that some functions do not have maxima: for example, <math>f(x) = x^2</math> takes on arbitrarily large values. A fundamental theorem states that any function defined on a [[compact space]] obtains a maximum value; this is just a consequence of the fact that the image of a compact space under a continuous function is compact, and thus bounded.
A simple example of this distinction is the following: one can define a function on the surface of the earth, whose value at a point is the current temperature at that point. One could similarly define such a function on the entire universe, sending a location to its temperature. Since the earth is compact, it follows that there is some point on earth which is the hottest at any given time. However, if the universe is unbounded, it is not compact, and we can not reach the same conclusion! There may be points in the universe which have arbitrarily large temperatures, so that the function has no maximum.
[[Category:calculus]]
[[Category:calculus]]
[[Category:mathematics]]
[[Category:mathematics]]