Fermat's theorem (stationary points)

Fermat's theorem is all about finding the highest or the lowest point on a big mountain (we call it a curve) with a special trick. Whenever we are on this mountain, we can look around us and see all the places where we could go either up or down.

Now, Fermat's theorem tells us that on this big mountain, there will always be a special spot that we call a stationary point. We can think of this spot like a big rock that just stays still and doesn't move up or down.

If we are standing right on top of this stationary point, we won't be able to go any higher or lower. It's like a magic spot where everything around it is either higher or lower than this point.

So why is this important? Well, sometimes we want to find the highest point or the lowest point on this mountain, and we don't want to spend all day climbing up and down. By using Fermat's theorem and finding the stationary point, we can quickly figure out whether we are standing on the highest or lowest point without actually climbing up and down the mountain.

In math, we call the stationary point the critical point. And we can use some fancy math tricks (called derivatives) to help us find these special spots on the mountain. This is how Fermat's theorem helps us solve some really complicated math problems without having to climb up and down a big mountain!