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Inflection points from graphs of function & derivatives

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.10 (EK)
,
FUN‑4.A.11 (EK)

Video transcript

what we're going to do in this video is try to get a graphical appreciation for inflection points which we also cover in some detail in other videos so the first thing to appreciate is an inflection point is a point on our graph where our slope goes from decreasing to increasing or from increasing to decreasing so right over here I have the graph of some function and let me draw the slope of a tangent line at different points so when X is equal to negative 2 that's it that is what the tangent line looks like and you can see its slope and then as we increase X we can see that the slope is positive but it is decreasing then it goes to 0 and then it goes negative and the slope keeps decreasing all the way until we get it looks like we get to about x equals negative 1 and then our slope begins to increase again so something interesting happened right at x equals negative 1 and so that's a pretty good indication we're just doing it graphically here we're not proving it but that at this point right over here we have an inflection point so let me write that down so let me show you that again now that we the point is labeled for exit negative 2 we have a positive slope it decreases decreases decreases it's negative it still decreases x equals negative 1 and then our slope begins increasing again so that's how you could tell it just from the function itself but you could also tell inflection points by looking at your first derivative remember an inflection point is when our slope goes from increasing to decreasing or from decreasing to increasing the derivative is just the slope of the tangent line so this right over here this is the derivative of our original blue function so here we can see the interesting parts and so notice what's happening on the derivative are the derivative is decreasing which means the slope of our tangent line of our original function is decreasing and we saw that notice while the derivative is decreasing right over here our slope will be decreasing our slope is positive our slope is positive but decreasing then it becomes negative but decreasing all the way until this point which is at x equals negative 1 so let's do that again so our slope is positive and decreasing and then right over about there right over here our slope keeps decreasing but then it actually turns negative and it keeps decreasing all the way until x equals negative 1 and then our slope begins increasing again so the derivative begins increasing which means the slope of our tangent line of our original function begins increasing so that point is interest interesting an inflection point one way to identify an inflection point from the first derivative is to look at a minimum point or to look at a maximum point because that shows a a place where your derivative is changing direction it's going from increasing to decreasing or in this case from decreasing to increasing which tells you that this is likely an inflection point now let's think about the second derivative so right over here this is the derivative of the derivative and I could zoom out to look at the whole thing you got you can't see the whole thing right over here actually I can zoom out a little bit more so that you can really see what's going on and so what's interesting here well it looks like right at x equals negative 1 we cross our second derivative crosses the x axis let me label that so right over there we cross the x axis which is exactly where we have the inflection point and that makes sense because our if our second derivative goes from being negative to positive that means our first derivative goes from being decreasing to increasing which means the slope of our tangent line of our function goes from decreasing to increasing we've seen that run over decreasing to increasing right over here now it's important realise the second derivative doesn't need to just touch the x-axis it needs to cross it so you might say well what about this point right over here 2 comma 0 the second derivative touches the x-axis there but it does it cross it so we never go from our derivative increasing to our derivative decreasing so big takeaways you can figure out the inflection point from you to the graph of the function from the graph of the derivative or the graph of the second derivative on the function itself you just want to inspect the slopes of the tangent line and think about where does it go from decreasing to increasing or the other way around from increasing to decreasing if you're looking at the first derivative you really just want to look at minimum or maximum points and if you're looking at the second derivative which we have in orange you want to look at what x value are we crossing the x-axis not just touching it but crossing the x-axis