Justifying properties of functions using the second derivative
Worked example: Inflection points from second derivative
- [Instructor] Let g be a twice differentiable function defined over the closed interval from negative seven to seven, so it includes those endpoints of the interval. This is the graph of its second derivative, g prime prime, so that's the graph right over there, y is equal to g prime prime of x. And they ask us how many inflection points does the graph of g have? So, let's just remind ourselves what an inflection point is. So, that is when we go from being concave downwards to concave upwards, so something like this. And another way to think about it, a point where our slope goes from decreasing to increasing. So, here our slope is that, then it's a little lower, then it's a little lower, then it's a little lower, but then all of a sudden it starts increasing again. It starts increasing, getting higher, higher, and higher. So, that would be an inflection point, whatever x value where that would actually happen. That would be the inflection point. You could go the other way around. You could have a function that looks something like this where we have a negative slope, but then our slope is increasing, slope is increasing, slope is increasing, but then our slope begins decreasing again. This too would be an inflection point. So, in other videos we go into more of the intuition of how do you think about the first and second derivatives of a function at an inflection point? But the big picture, at least for the purposes of this worked example, is to realize when you're looking at the second derivative, you have an inflection point where the second derivative crosses the x axis. It's not enough to just touch the x axis. You must cross the x axis. And so right over here we are crossing the x axis, so that is an inflection point. Right over here, we are crossing the x axis, so that is an inflection point. Here and here we touch the x axis. Our second derivative is equal to zero. But we don't cross. We don't cross the actual x axis. We don't go from being positive to negative. We stay non-negative this entire time. Similarly right over here, maybe something interesting happens past this point, but they're telling us that the function is only defined over this interval, so actually nothing happens beyond getting that point, so we're not going to cross the x axis. So, to answer the question, how many inflection points does the graph g have, well, it has two inflection points looking at the second derivative here. Now, we know the answer. Why does that make sense? Why do you have to cross the x axis? Well, let's just imagine. Let's say that this is the graph of a second derivative. So, this is f prime prime. So, the first derivative, for example, could look like this. The first derivative might look like this, where over here we have a negative slope, negative slope, negative slope, negative slope, but it's getting closer and closer, and then right over here, all of a sudden the slope becomes positive and increasing. So, that would be f prime of x. And then you could think about, well, if this is describing the derivative of our function, then what's our function going to look like? Well, our function over here would have a very positive slope, but then the slope would keep decreasing all the way up until this point, and then it increases again. So, we have a positive slope right over here. So, for example, our function might look like this. It might have a very positive slope, but then the slope keeps decreasing, and then right over here all of a sudden, the slope begins increasing. The slope begins increasing again. And so here we were concave downward over this first part, over this first part. We have a positive slope, but it's decreasing. Positive slope, but it's decreasing. And then we go to having a positive slope, but now we are increasing again. And so this should give you a good sense for why you need to cross the x axis in the second derivative.