Are the words maximum/minimum used interchangeably with the words maxima/minima? I have been learning the calculus of maxima and minima recently and have seen both types of words 
maximum minimum
maxima minima
It appears that the words are used in the same sense in both forms, but I do not know that for sure. If this is true, why do we have two words for the same thing? If it is not true, what is the distinction between the meanings of these two words?
 A: Maxima and minima are the plural forms of maximum and minimum respectively.
So when a function has more than one (local) maximum, it makes sense to talk about its maxima. And so forth.
A: As noted in Deepak's answer, "maxima" and "minima" are the plural forms of "maximum" and "minimum", respectively.  The words "maximum" and "minimum" are derived from Latin.  Latin derived words ending in "-um" are made plural by replacing the suffix "-um" with "-a".  For example:
\begin{array}{ll}
\hline\textbf{Singular} & \textbf{Plural} \\\hline
\text{maximum} & \text{maxima} \\
\text{minimum} & \text{minima} \\
\text{extremum} & \text{extrema} \\
\text{optimum} & \text{optima}^{[1]} \\
\text{datum} & \text{data}^{[2]} \\\hline
\end{array}
It may be worth noting that the more English-standard pluralizations "maximums" and "minimums" are acceptable, though used less frequently.  My guess is that, over time, "maximums" and "minimums" will eventually replace the standard Latinate plurals.  Google's ngram viewer is a good way of seeing how the words are used through time:


[1] The Nissan Optima makes me sad... how can one car be plural?
[2] The word "data" is, in fact, plural.  However, in English, this battle seems to have been lost, as most people use "data" as a collective noun. Language changes over time.  C'est la vie. 
