In maxima problems you are trying to maximize something. A typical problem involves a farmer with 100 feet of fence who wants to enclose as much area as possible. He decides to further maximize the enclosure by using the side of his barn for one side of the enclosure and the fence for the other three sides. He needs to know how much of the fence should be parallel to the barn. We start by expressing the area (what we want to maximize) in terms of the variable we can control (the part of the fence parallel to the barn). Area = length x width = P x 1/2(100 - P). The derivative of the area will be 50 - P. If 50 - P = 0, then P = 50. If 50 feet of fence runs parallel to the barn, the area will be maximized. If P = 50, the area is 50 x 25 = 1250. If P is a little longer, the area is 52 x 24 = 1248. If P is a little shorter, the area is 48 x 26 = 1248. Clearly, P = 50 gives a maximum area.
Minima problems involve equations where we want to minimize something. Examples include finding the shortest ladder that can go over a 10-foot fence to rest on a wall two feet behind the fence. Express the length of the ladder in terms of some other variable -- such as the distance from the fence to the foot of the ladder -- then find the derivative of the formula and set it to zero. Solve the resulting equation to get the distance from the foot of the ladder to the fence, then use that to find the length of the ladder. Another example is to find the minimum cost of making a cylindrical tin can that holds 24 cubic inches of something. Use the formula for the volume of a cylinder to express the height in terms of the radius, then use this to express the formula for the surface area of a cylinder, differentiate and set to zero and solve.
The general solution is to express the quantity you want to maximize or minimize in terms of another variable. Then differentiate the formula and set it to be equal to zero. Then solve this new equation to get the value of this variable that will make the formula a maximum or a minimum. Calculus even offers a way to tell if the answer you got was a maximum or a minimum. Take the derivative of the derivative (called the double derivative). If the double derivative is negative, you have found a maximum value. If the double derivative is positive, you have found a minimum.