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## Concavity and inflection points intro

# Inflection points (graphical)

AP.CALC:

FUN‑4 (EU)

, FUN‑4.A (LO)

, FUN‑4.A.4 (EK)

, FUN‑4.A.5 (EK)

## Video transcript

- [Voiceover] We're told let g be a differentiable function defined over the closed interval
from negative four to four. 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? So inflection points are
where we change concavity. So we go from concave, concave upwards, upwards,
actually let me just draw it graphically. We're going from concave
upwards to concave downwards, or concave downwards to concave upwards. Or another way you could think about it, you could say we're going
from our slope increasing, increasing, increasing, to our slope decreasing. To our slope decreasing, or the other way around. Any points where your
slope goes from decreasing, our slope goes from
decreasing, to increasing. To increasing. So let's think about that. So as we start off right over here, at the extreme left it's 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, it's getting a little bit flatter, so our slope is at a very high
level but it's decreasing. It's decreasing, decreasing, decreasing, slope is decreasing, decreasing even more. It's even more and then
it's actually going to zero, our slope is zero and
then it becomes negative. So our slope is still decreasing. Then it's becoming more
and more and more negative, and then right around, and
then right around here, it looks like it starts
becoming less negative, or starts increasing. So our slope is increasing increasing, it's really just becoming
less and less negative, and 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. So it looks like our slope is decreasing. It's becoming more and more
and more and more negative, and so it looks like
something interesting happened right over there, we
got a transition point, and 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
and 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, and so slope increasing was
here to slope decreasing, so 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 that we can see on this graph? Well it has three over the interval that at least we can see.