Finding increasing interval given the derivative

- [Voiceover] Let g be a function defined for all real numbers. Also let g prime, the derivative of g, be defined as g prime of x is equal to x squared over x minus two to the third power. On which intervals is g increasing? Well, at first you might say, they don't even give us g. How do we figure out when g is increasing? Well, the answer is all we need is g prime which they do give us. And saying on which intervals is g increasing, that's equivalent to saying, on which intervals is the first derivative with respect to x on which intervals is that going to be greater than zero? If your rate of change with respect to x is greater than zero, if it's positive, then your function itself is going to be increasing. And so there's a couple of ways that we could approach this. You might just want to inspect kind of the structure of this expression and think about, well, when is that going to be greater than zero or we could do it a little bit more methodically. We could say, well, let's look at the critical points or the critical values for g. So critical, critical points for g and just to remind ourselves what critical points are, that is when g prime of x is equal to zero or g prime of x is undefined, is undefined, and we have videos on critical points or critical values and why those are relevant is those are the places, those are possible places where the sign could change, the sign of g prime could change. So when is g prime of x equal to zero? Well, the way to get g prime of x equal to zero is getting the numerator equal to zero and that's only going to happen if x squared is equal to zero or if x is equal to zero. So that's the only place where g prime of x is equal to zero and where is g prime of x undefined? Well, it's going to be undefined if the denominator becomes undefined. The denominator becomes undefined if the denominator is zero and so that's going to happen if x minus two is equal to zero, x minus two is equal to zero or x is equal to two. So we have two critical points or critical values here and what I'm going to do is I want to graph them. Let's put them on a number line and let's just think about what g prime is doing in the intervals between the critical points. So let's start at zero, one, two, three and then let's go to negative one and we have a critical point at, let me do that in magenta, we have a critical point at x equals zero right over there and we have a critical point at x equals, at x equals two right over there. And so let's think about what g prime is doing in the intervals between the critical values or on either side of the critical values. So let's think about, let's first think about this interval. Let me do it in this purple color. Let's think about the interval between, between negative infinity and zero. So if we think about this interval, so negative infinity and zero, that open interval, well, if we look at g prime, the numerator is still going to be positive. If you take any negative value squared, you're going to get a positive value so this is going to be positive. Now, what about the denominator? You take a negative number, you subtract two from it, you're still going to get a negative number and then you take it to the third power. Well, a negative number to the third power is going to be a negative number so that right over there is going to be negative. So you're going to have a positive divided by a negative so g prime is going to be negative so let me write that down. So on this interval, on this interval, I'll write it like this. g prime of x is less than zero or if we cared or if we want to know when it's decreasing, we would know it's definitely decreasing over that interval. Now, let's take the interval between zero and two right over here. So this is the interval from zero to two, the open interval. So what's going to go on with g prime of x here? Well, once again, x squared, anything greater than a zero and it says we're not including zero in this interval. Well, this is for sure going to be positive and so let's see, if we have x minus two where x is greater than zero but less than two. So if x, we could just say for example, if x was one, one minus two is negative one. We're still going to get negative values in this denominator right over here. So since we're still going to get negative values in this denominator, the denominator is still going to be, you take a negative value to the third power, well, you're going to still get a negative value so this is going to be negative. So you're still going to have g prime as less than zero so let me write that down. So you still have g prime of x is less than zero. And then let's take the interval above. Let's take the interval from two to infinity. Two to infinity. Well, the numerator is positive. It's always going to be positive for any x not being equal to zero and this denominator, you're taking values greater than two, subtracting two from it which is still going to give you a positive value. You take the third power, it's all going to be positive. It is all going to be positive. So this is the interval where g prime of x is greater than zero. So on which intervals is g increasing? Well, that's where g prime of x is greater than zero so it's going to be from two, from two to infinity or we could just write it like this. We could write x is greater than two. Either way, for either of these, g prime of x is greater than zero and your function g is going to be increasing.