Intervals where a function is positive, negative, increasing, or decreasing
- [Voiceover] What I hope to do in this video is look at this graph y is equal to f of x and think about the intervals where this graph is positive or negative and then think about the intervals when this graph is increasing or decreasing. So first let's just think about when is this function, when is this function positive? Well positive means that the value of the function is greater than zero. It means that the value of the function this means that the function is sitting above the x-axis. So it's sitting above the x-axis in this place right over here that I am highlighting in yellow and it is also sitting above the x-axis over here. And if we wanted to, if we wanted to write those intervals mathematically. Well let's see, let's say that this point, let's say that this point right over here is x equals a. Let's say that this right over here is x equals b and this right over here is x equals c. Then it's positive, it's positive as long as x is between a and b. At x equals a or at x equals b the value of our function is zero but it's positive when x is between a and b, a and b or if x is greater than c. X is, we could write it there, c is less than x or we could write that x is greater than c. These are the intervals when our function is positive. Let me write this, f of x, f of x positive when x is in this interval or this interval or that interval. So when is f of x negative? Let me do this in another color. F of x is going to be negative. Well, it's gonna be negative if x is less than a. So this is if x is less than a or if x is between b and c then we see that f of x is below the x-axis. F of x is down here so this is where it's negative. So here or, or x is between b or c, x is between b and c. And I'm not saying less than or equal to because at b or c the value of the function f of b is zero, f of c is zero. That's where we are actually intersecting the x-axis. So that was reasonably straightforward. Now let's ask ourselves a different question. When is the function increasing or decreasing? So when is f of x, f of x increasing? Well increasing, one way to think about it is every time that x is increasing then y should be increasing or another way to think about it, you have a, you have a positive rate of change of y with respect to x. We could even think about it as imagine if you had a tangent line at any of these points. If you had a tangent line at any of these points the slope of that tangent line is going to be positive. But the easiest way for me to think about it is as you increase x you're going to be increasing y. So where is the function increasing? Well I'm doing it in blue. So it's increasing right until we get to this point right over here, right until we get to that point over there then it starts decreasing until we get to this point right over here and then it starts increasing again. It starts, it starts increasing again. So let me make some more labels here. So let's say that this, this is x equals d and that this right over here, actually let me do that in green color, so let's say this is x equals d. Now it's not a, d, b but you get the picture and let's say that this is x is equal to, x is equal to, let me redo it a little bit, x is equal to e. X is equal to e. So when is this function increasing? Well it's increasing if x is less than d, x is less than d and I'm not gonna say less than or equal to 'cause right at x equals d it looks like just for that moment the slope of the tangent line looks like it would be, it would be constant. We're going from increasing to decreasing so right at d we're neither increasing or decreasing. But then we're also increasing, so if x is less than d or x is greater than e, or x is greater than e. And where is f of x decreasing? So f of x, let me do this in a different color. When is, let me pick a mauve, so f of x decreasing, decreasing well it's going to be right over here. It's gonna be right between d and e. Between x equals d and x equals e but not exactly at those points 'cause at both of those points you're neither increasing nor decreasing but you see right over here as x increases, as you increase your x what's happening to your y? If you go from this point and you increase your x what happened to your y? Your y has decreased. You increase your x, your y has decreased, you increase your x, y has decreased, increase x, y has decreased all the way until this point over here. So f of x is decreasing for x between d and e. So hopefully that gives you a sense of things. Notice, these aren't the same intervals. That we are, the intervals where we're positive or negative don't perfectly coincide with when we are increasing or decreasing. So it's very important to think about these separately even though they kinda sound the same.