What are absolute and relative extremes?

All of us have already gone through ups and downs in our lives. There are times when it seems that the world has joined together to treat us badly and so everything goes wrong. Fortunately, there are moments when the opposite takes place and we feel very happy. Those ups and downs often happen in our daily lives. As Maths is often used to solve these real problems, there are lots of tools to detect these moments. So, as to be able to answer the question we have been asked, we will make use of a simple example:

To distinguish absolute extremes from relative extremes is quite simple. An absolute extreme is just the place where the function reaches its higher point or its lower point. In the previous graph we can notice that the point `a`, is the highest one and it is called absolute maximum. On the other hand, the point `x_5` shows us the lowest point, which is called the absolute minimum. However, there are other points where the function reaches a minimum placement; for instance, the point `x_2` is the lowest in that “neighbourhood”. So, it is called relative minimum. Besides that, the point `x_6` is the highest one next to its neighbours and therefore it is called the relative maximum. A function can have several relative maxima and minima, but it is obvious it can only have an absolute maximum and an absolute minimum.

What is that useful for?

Let’s suppose the previous graph shows the sales of an enterprise during a year. So, the horizontal axis represents the passage of months and the vertical axis (which is not visible) represents the volume of sales. In this situation it is important to know when the sales reach a maximum and a minimum peak. After knowing about that we can order the production sector to intensify work if we predict a high point in sales. On the contrary, you can do without some workers or you can put them doing other tasks if you previously know that you are about significantly decreasing your sales. There are several optimization problems and so it is necessary to find the place where the function reaches a maximum or a minimum. It can both happen to find the maximum profit of an enterprise or to find the minimum expense to produce a certain product. Mathematics allows us to efficiently answer to all those questions!

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