Connecting a function, its first derivative, and its second derivative
Inflection points from graphs of function & derivatives
- [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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