 Hi, everyone. Welcome to this lesson on using the first derivative. This lesson delves into how derivatives can be used to determine relative extrema of a function, as well as the intervals on which the function is increasing and decreasing. If you take a look at the graph that you see here, as a point moves along the curve from A to B, the function values, the y values, increase as the x value increases. As a point moves from point B to point C, the function values decrease as the x values increase. We say then that the function f is increasing on the interval from x sub 1 to x sub 2 and decreasing on the interval from x sub 2 to x sub 3. Notice how we use the x values to identify the interval on which the function is increasing or decreasing. So how does this relate to derivatives then? Let's consider on the curve locations at which the tangent line to the curve would have a positive slope. Now, there's infinitely many places at which that would occur because there's infinitely many points on any interval. So for example, if I had a point here on the curve in between the point A and the point B, if I were to draw a tangent line, that tangent has a positive slope. Same thing up here. Tangent lines here always have a positive slope. If I consider the interval in between point C and point D, same thing. I can pick a point and the tangent there has a positive slope. And then same would be true, it looks like, from E to F and then F to G and so on. If you think about the common characteristics here, every place I was able to draw a tangent line that has a positive slope, the curve, the function, is increasing at that location. So we can conclude that if the tangent line has a positive slope, that means it has a positive derivative. Because remember, the derivative is defined to be the slope of the tangent line to the curve. And it's on those intervals at those locations that the function is increasing. Similarly, let's now consider the locations on the curve at which the tangent line is going to have a negative slope. Again, there's infinitely many places. So let's start here in between B and C. So if I drew a point here, tangent to the curve there has a negative slope. Same thing down here. The next interval that looks like it is decreasing is in between D and D. And once again, if I pick a point and draw a tangent line, that tangent line has a negative slope. So once again, think about the common characteristics. Now if the tangent line had a negative slope to it, that tells us the derivative therefore will be negative. And it's at those locations that the function is decreasing. Finally, let's consider the locations on the curve at which the tangent has a slope of 0. Now you'll see these are already drawn in for us. There's one here at B, one at C, one at E, and one at F. Notice D does not have one. That is a cusp point. And remember, there is no tangent that can be drawn at a cusp point. So if you consider points B, C, and E, it's at these locations the tangent line has a slope of 0, meaning it's a horizontal tangent line. Therefore, where the tangent slope is 0, the derivative is 0, and the function at those locations is constant. It is not changing. So what can we do with this information then? It leads us to the test for increasing and decreasing functions. If F is continuous on the closed interval from A to B and differentiable on the open interval from A to B, if the derivative is positive greater than 0 for all x's in that open interval from A to B, then F is increasing on the interval from A to B. If the derivative is negative less than 0, then for all x's in the interval from A to B, F is going to be decreasing on the interval from A to B. Finally, if the derivative equals 0 for all x's in the open interval from A to B, then F is constant on the interval from A to B. So how then will we determine where a function is actually increasing or decreasing? Well, the first thing we'll have to do is find the derivative. Secondly, we will determine the critical numbers of the function, the values of x for which F prime of x equals 0 or F prime of x does not exist. Third, we will determine the sign of the derivative on the intervals formed by the critical numbers. And we're going to do this by way of a number line analysis. Finally, once we have our number line analysis, we are going to apply the test for increasing and decreasing functions. Now once we know the intervals on which a function is increasing or decreasing, it is easy then to determine the relative extrema, the relative maximum points of the function. And the way we're going to do this is with the first derivative test for relative extrema. Suppose function F is continuous on the open interval from A to B that contains a specific x value that we will call C. And suppose F is differentiable on the open interval from A to B, except possibly at C. If the derivative F prime of x changes from negative to positive at C, then F has a relative minimum at C. Now think about why that makes sense. If F prime is changing from negative to positive, we know that tells us that this function is going to be changing from decreasing because that's where the derivative is negative to increasing because that's where the derivative is positive. And notice at the bottom you have a minimum that's formed. If F prime changes from positive to negative at C, then we have a relative maximum at C. Again, it makes sense because if F prime is changing from positive, meaning the function's increasing, to the derivative being negative, meaning the function F is decreasing, at the top we have a relative maximum created.