Inflection points (graphical) | AP Calculus AB | Khan Academy

We're told let G be a differentiable function defined over the closed interval from 4 to 4. The graph of G is given right over here, given below. How many inflection points does the graph of G have?

So let's just remind ourselves what are inflection points. Inflection points are where we change concavity.

So we go from concave upwards to concave downwards or concave downwards to concave upwards.

Another way you could think about it is that we're going from our slope increasing to our slope decreasing, or the other way around. Any points where your slope goes from decreasing to increasing.

So let's think about that. As we start off right over here, at the extreme left, it seems like we have a very high slope. It's a very steep curve, and then it stays increasing, but it's getting less positive.

So it's getting a little bit flatter. Our slope is at a very high level, but it's decreasing, decreasing, decreasing. The slope is increasing, decreasing even more, it's even more.

Then it’s actually going to zero; our slope is zero, and then it becomes negative. So our slope is still decreasing, and then it's becoming more and more negative.

Then right around here, it looks like it starts becoming less negative, or it starts increasing. So our slope is increasing; it's really just becoming less and less negative.

Then it’s going close to zero, approaching zero. It looks like our slope is zero right over here, but then it looks like right over there our slope begins decreasing again.

So it looks like our slope is decreasing again; it’s becoming more and more negative. It seems like something interesting happened right over there; we had a transition point.

Then right around here, it looks like it starts; the slope starts increasing again. So it looks like the slope starts increasing; it's negative, but it's becoming less and less and less negative.

Then it becomes zero, and then it becomes positive, and then more and more and more and more positive. So, inflection points are where we go from slope increasing to slope decreasing, so concave upwards to concave downwards.

This was an inflection point, and also from slope decreasing to slope increasing. So that's slope decreasing to slope increasing, and this is also slope decreasing to slope increasing.

So how many inflection points does the graph of G have? We can see that we've on this graph, well, it has three over the interval that at least we can see.