I said I would do a video on inflection point so let’s get intuition on inflection points. So let me draw a graph and let say, let me try a little bit neither that I did in the last video I think I rush to that a little bit more that I want to. So let’s say that that is the y-axis, that’s the x-axis so let say my graph look something like this. So it comes down like that in it and it kind of flattens out there and we said that inflection point are another example when the derivative equal zero. I don’t know if I drew it properly here. I want this to flatten out a little bit but you can take my word for that at this point right here the slope is flat. Maybe the graphs looks a little bit more that.
If I were to draw a line that represents a slope at that point it would look something like this. It would look something like this. All right, I’m trying my best. I know you get the point right. There are three types of places where you can have a zero slope, a minimum, maxima or an inflection point. So what will the first derivative look like at this point or around this point? Let’s draw a chart and see if we can get the intuition. A little bit scatter brain today that’s why I record the video and I added wrong and see.
So this is f of (x). Let’s see if we can sketch out what f prime of (x) or what the derivative of this function it looks like and I’m going to do down here just because it we can kind of follow along the same axis. Let me draw the x and y axis. So let’s say this is the y-axis again and let’s say that’s the x-axis and I want to graph f prime of (x) and actually let me erase a little bit of this. Let me erase this and I want to get this to get in the way but we care around at this point. So, what is the slope doing as we approach that point? Well here the slope is really negative and say at that point and then as we move arbitrarily to the right it becomes a little bit less negative, then here it becomes even less negative until finally it becomes zero.
So if we’re going to draw f prime of (x) at this point we said that the slope was really negative and then it becomes a little bit less negative, its getting less, less, less negative and the approaches a zero slope right there. But then what is it doing right now? In the previous example when we saw a concave upwards graph maybe the parabola we saw that you know that the slope would just kept increasing. It’s kind of the slope just kept increasing you go up that way but in this situation the slope becomes zero. It keeps becoming less negative up to zero but then what happens as you go here? It starts, the slope starts decreasing again. It’s a little bit negative there and then it becomes more negative and then it becomes more negative so the graph of the slope will actually look something like that. Right starts off really negative becomes less and less negative, becomes less and less negative. Zero at the point of inflection. That’s the slope at the point of inflection and then it becomes more negative again.
So what will the second derivative of that look like? Let me draw that here. I can even draw that on top of that same graph and maybe it’s easier for it to draw here. Well the second derivative looks like. Well the second derivative is just a slope of the slope right so it’s a slope of this line. So here we have a very high slope, the slope is very positive there then it becomes a little bit less positive and then it becomes a little bit even less positive until it becomes flat here. And then it becomes negative and more negative and more negative.
So if this point, the second derivative will look something like this. Back here right at this value of (x), let’s say that this is the value of (x) we have a really high slope and the slope just becomes a little bit flatter, a little bit flatter until I get to zero then its starts going negative. So the second derivative is going to look like this. It doesn’t have to be aligned but I’m kind of, this is a typical example you might see. So this would be f prime, prime of (x).
Now what can we now take away from this? Well you learn your concavity rules and all of that, you learn that if the second derivative is positive your concave upwards so f prime, prime at you know whatever point at some point A let say (x) is equal to A. Well first of all we know that if f prime of A is equal to zero we know the slope is zero there right and that’s what this for is on to represent it and so that tells us that its either an inflection point, a maxima point or a minimum point right. And then we learned in the previous video that if your second derivative at A is greater than zero then your concave upward and you might want to review that if you actually just draw then think about if for yourself. I don’t want you take any of this for granted. It’s the value of learning this is to get the intuition not to memorize this rules. And we also learned that if f prime, prime of A is negative that we’re going concave downwards and now this example; what is f prime if this was A right here, what’s f prime, prime A in this example? Well this is A right here. Well it’s a zero right because if you see the graph of the slope flattens out here, it starts going negative again here.
So well if the first derivative point is zero it’s either maximum, a minimum or an inflection point and if we know that if the second derivative is also zero that is definitely an inflection point. So really that’s the intuition behind all of you know the maximum and minimum problems that you’ll see in your first semester calculus course and now I hope you have the intuition and we can do some problems. See you in the next video.
Transcription by:
Scribe4you Transcription Services