Maximum and minimum

Imagine you are walking along a winding, bumpy path.

You might climb a small hill that is higher than the ground right around you. In math, we call this a local maximum. You might also step down into a small valley. This low point is called a local minimum. These high and low points are known together as extrema. Sometimes, a hill is the highest point on the entire path. This is called a global maximum. A valley might also be the deepest point of all, which is a global minimum.

Finding these points is a very important job called mathematical optimization. To find a global maximum, you can look at all the local peaks in the middle of your path. You must also check the very edges or boundaries of the path. The highest of all those points is your global maximum.

For certain types of smooth functions, there is a special rule. This rule says that local peaks must happen at points where the slope is zero. These are called critical points. However, not every critical point is a peak or a valley. You can use special tests to see if a point is a maximum or a minimum.

Mathematicians have studied these ideas for a very long time. One of the first people to suggest a way to find these points was Pierre de Fermat. He used a technique called adequality to find the highs and lows of functions. Later, other mathematicians used tools like the extreme value theorem. This theorem says that if a path is continuous and stays within certain bounds, it must have a highest and lowest point. This helps us know for sure that a maximum and minimum exist before we even start looking for them.

Math shows us many different ways these points can appear. For example, a function like x squared has only one unique global minimum at zero. Another function, like x cubed, has no global maximum or minimum at all. Some functions, like a cosine wave, have infinitely many global peaks and valleys.

Even simple shapes can have these points. If you have a certain amount of fencing, you can use math to find the best way to make a rectangle. This helps you find the maximum area you can enclose with your fence.

We can also find these points in groups of numbers called sets. In a set, the greatest element is the maximum. If a set of numbers goes on forever, like the natural numbers, it might not have a maximum. The natural numbers do have a minimum, but they never end.

In more complex math, we can look at shapes in many dimensions. A point might look like a minimum locally, but a much deeper valley could exist elsewhere. This shows how important it is to check the whole area, not just the spot where you are standing.