And and why is this concave upwards?

We know it's concave upwards.

So you have concave, and you have convex.

It looks like this side has been caved in.

Now, in concave downwards, the slope is decreasing.

So, for example, this is a concave quadrilateral

I want to show you what concave downwards looks like.

This point right here, where we go from concave downwards to concave upwards, and it would actually be true if it was the other way around, this is called an inflection point.

It has to go from being negative in a concave downward to positive in concave upwards, or it has to go from positive in concave upwards to negative in concave downwards.

And similarly, you know that it's concave upwards when the second derivative is positive.

And we know that we're dealing with, the function is concave upwards at this point.

And one way to define concave quadrilaterals, so let me draw it a little bit bigger, so this right over here is a concave quadrilateral, is that it has an interior angle, it has an interior angle that is larger than 180 degrees.

So a very simple way to think about it is that concave upwards is kind of a U shape.

So that tells us that not only are we monotonically decreasing as we approach 3, but we're concave upwards.

So what did we say about concave upwards? concave upwards is when the slope is increasing, which means that the rate of change of the slope is positive.

So up until this point, the slope is decreasing, which is completely consistent with what we said about being concave downwards.

Hopefully, you understand visually what concave upwards and concave downwards means, and an inflection point, but what does this mean for the second derivative?

Now, over what interval of this function's domain, maybe I'll call it h of x, is this curve concave upwards or concave downwards?

Now that we know the second derivative, we can answer the question, is my graph concave upwards or downwards at either of these points?

So the two ideas that we'll talk about, one is being concave upwards I'll write that in this column right here, and maybe in pink I'll show you what the concave downwards looks like.

Well, what I want to do in this video is, one, show you what concativity is, or what concave upwards and what concave downwards is, and to have an intuition of what those mean, and then discuss the ideas of inflection points, which are really just transition points between being either concave upwards and downwards or between downwards and upwards.

If we went from a curve let me draw a curve that looks like this that was concave upwards and then it's concave downwards, the point that we're switching, the point at which you go from one to the other, essentially your rate of the change of slope switches signs, that is an inflection.

In 1876 a French engineer, A. Mangin, invented what has come to be called the Mangin mirror, a concave glass reflector with the silver surface on the rear side of the glass.

It's an interesting proof, maybe I'll do a video, it's actually a pretty simple proof, to show that if you have a concave quadrilateral if at least one of the interior angles has a measure larger than 180 degrees that none of the sides can be parallel to each other.