Difference between maximum and minimum? If I have a problem such as this: 
We need to enclose a field with a fence.  We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing.  Determine the dimensions of the field that will enclose the largest area.
This is obviously a maximizing question, however what would I do differently if I needed to minimize the area? Whenever I do these problems I just take the derivative, make it equal to zero and find x, and the answer always seems to appear - but what does one do differently to find the minimum or maximum?

The problem of minimizing and the problem of maximizing are essentially the same.
As you say, take the derivative of your function and find the critical points. Do not forget, however, that global extrema can also occur at endpoints of the reasonable domain (i.e. the values of $x$ allowed within the context).
For the fence problem, suppose we let $x$ represent the width of the fence. You would first write a function for area in terms of $x$. Notice that this function, although technically defined for all reals, really only makes sense on $0 \leq x \leq 500$ , as the width of the fence cannot be smaller than 0 nor larger than 500. Thus, you need to check not only whatever critical points you find, but also $x = 0$ and $x = 500$. To determine which is the maximum and which is the minimum, you simply compare the values of the function at all these interesting points.
