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## Connecting a function, its first derivative, and its second derivative

# 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

- [Instructor] 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 two, 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 zero, 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 one, and then our slope
begins to increase again. So, something interesting happened right at x equals negative one, 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 the point is labeled. For x at negative two,
we have a positive slope. It decreases, decreases, decreases. It's negative, it still
decreases, x equals negative one, 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, 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 one. 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 one, 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 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 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 actually 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 one, we cross, our second
derivative crosses the x axis, so 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 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 over and over, decreasing to increasing right over here. Now, it's important to realize 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, two, zero?" The second derivative
touches the x axis there, but it doesn't cross it, so we never go from our
derivative increasing to our derivative decreasing. So, big takeaways, you could figure out the inflection point from either the graph of the function, from the graph of the derivative, or the graph of the second derivative. On the function itself, you just wanna 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 wanna look
at minimum or maximum points. And if you're looking at
the second derivative, which have we in orange, you wanna look at at what x value are we
crossing the x axis? Not just touching it,
but crossing the x axis.

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