If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Inflection points (graphical)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.4 (EK)
,
FUN‑4.A.5 (EK)

## Video transcript

we're told let G be a differentiable function defined over the closed interval from negative 4 to 4 the graph of G is given right over here given below how many inflection points does the graph of G have and so let's just remind ourselves what our 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 so 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 so we 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 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 as 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 it starts increasing so our slope is increasing increasing it's really just becoming less and less negative and then it's be 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 becoming more and more and more and more negative and so looks like something interesting happened right over there we had 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 we cut but it's becoming less and less and less negative and then it becomes 0 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 the slope decreasing so this was an inflection point and also from slope decreasing the slope increasing so that's slope decreasing the 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 that we on this graph well it has it has three over the interval that at least week we can see
AP® is a registered trademark of the College Board, which has not reviewed this resource.