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?

When you take the derivative of a function $f(x)$ and solve for where it is zero you are finding the points at which the function does not change. This includes maxima and minima. The fact that it always seems to work for you leads me to believe that you don't really know how it works.
When you find the extreme points you need to perform the second derivative or some other test to make sure that what you get is a maximum or a minimum.
So, my answer is that in both cases you do basically the same thing, except in the end where you pick your maximum or minimum. For a concrete example, consider the function $$f(x) = 2x^3-3x^2.$$
We shall find the local minimum and local maximum. When you take the derivative of $f(x)$ you obtain $$f'(x) = 6x^2-6x = 6x(x-1).$$ You see that the function $f(x)$ has two extreme points - one at $x=0$ and one at $x=1$. We take the second derivative to perform the second derivative test $$f''(x) = 12x-6.$$ We see that $f''(0) = -6 <0$ so the function $f(x)$ is concave down, meaning $f(0)$ is a maximum. Also $f''(1) = 6 > 0$, so the function is concave up, meaning that $f(1)$ is a minimum. Note that if $f''(x) = 0$ at some point, the second derivative test gives us no information.
