 Hi there, and welcome to the screencast about critical values. In this screencast we're going to try to identify the critical values of a function given only the graph of the function and without the formula. So here's the graph of a function. Before we do anything, I'd like to have you do something, and that is pause the video and think about this graph, look at it, and write down how many critical values does this function have. So go do that while you pause the video, and then when you're done, write down your answer, unpause the video, and we'll talk about it. So the answer is that this function has six critical values on it. Now let's find all six of those. And to do that we're going to need to first recall the definition of a critical value. Remember that a critical value for a function is any point in the domain of that function that makes its derivative either zero or undefined. So let's look through the graph and see if we can find in all those places. Well, let's first of all think about where the derivative is equal to zero. Now the derivative tells us the slope of the tangent line. So three places that jump out at you immediately, or where the slope of the tangent line is zero, and that will be here, here, and here. All those places have a horizontal tangent line if you drew it to the graph like so. So there's three of the critical values right off the bat that are quite easy to locate. Now the fourth place on, there's a fourth place on this graph where the derivative is also equal to zero. It's always obvious, and it's right here. At this point if I were to draw the graph of, or draw the tangent line to the graph of this function, the graph is locally linear right there, so it does have a tangent line. It would look like this. It would sort of cut through the graph right there, but it would still be considered a tangent line. And as you can see that's a horizontal tangent line. That's a shelf point. It's a place that has a derivative equal to zero. It's momentarily horizontal, but it's neither a local maximum nor a local minimum. Despite that, that is still a critical value because the derivative is zero there. So there's four places right off the bat out of the six that give us a derivative equal to zero, and that makes some critical values. So where are the other two? Well, that's where we need to look at the second part of the definition of a critical value, and that is where the derivative is undefined. Now the derivative being undefined means that your function is not differentiable. I claim that this function is not differentiable at two different points. Let's point out where they are. One place where the function is not differentiable is right here at this cusp. The function is not differentiable there because if I were to zoom in on that cusp, what I would see is a sharp point no matter how close in I zoomed. So the function at this point is not locally linear. It makes its derivative undefined, and therefore that point I put in the box right there is a critical value. There's one other place, the sixth place, where the derivative fails to be defined, and it's maybe the hardest of all to see it, it's right here. Now what's happening here, the graph is locally linear, so it seems like you ought to have a defined derivative, but if you look carefully at it, zooming in on this point would show a vertical tangent line. So that tangent line, if I drew it to the point right where I drew the box, would be a vertical tangent line. The slope of a vertical tangent line is undefined. Therefore the derivative is undefined there as well, and that makes that point a critical value. So there are your six points, right here, here, here, here, here, and here. And all those places the derivative is either zero or for various reasons undefined. Before we leave the video I just want to point out that not all of these critical values led to local extreme values. We only had four local extreme values, that's this one, this one, this one, and this one as well as a local maximum. It's actually the absolute maximum of this function. So there was a critical value here and a critical value here that did not lead to extreme values. So let's just note that the number of extreme values, local extreme values of any given function is typically not equal to the number of critical values, but less than or equal to the number of critical values. Not every critical value, especially if you look at this one right here, is pretty clear, leads to a local extreme value. Thanks for watching.