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Worked example: finding relative extrema

Sal finds the relative maximum point of g(x)=x⁴-x⁵ by analyzing the intervals where its derivative, g', is negative or positive.

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Video transcript

- [Voiceover] So we have g(x) being equal to x to the fourth minus x to the fifth, and what we wanna do without having to graph g, we want to figure out at what x values does g have a relative maximum? And just to remind us what's going on in a relative maximum, so let me draw a hypothetical function right over here, so a relative maximum is going to happen, so you can visually inspect this, okay that looks like a relative maximum. That's kind of a top of a mountain or top of a hill, these all look like relative maximum and what's in common? Well the graph, the function is going from increasing to decreasing at each of those points. It's going from increasing to decreasing. Increasing to decreasing at either of the points, or you could say that the first derivative is going from positive to negative. So if you look at this interval right over here, g prime is greater than zero, and then over the next interval when you're decreasing, g prime would be less than zero. So what we really need to think about is when does g prime, so let me see, relative, we care about relative maximum point, and so that's essentially asking when does g prime go from positive to negative? From, from, I wrote fror. From g prime greater than zero, to g prime less than zero, and the values that we could look at or the points are our critical points and critical points are where g prime is either zero or it is undefined. So let's think about it, where is g prime of x equal to zero? g prime of x is equal to zero when well let's just take g prime of x. We're gonna leverage the power rule right here, four x to the third power, four x to the third, minus five x to the fourth. Minus five x to the fourth is equal to zero. Let's see, we can factor out an x to the third. So we have x to the third times four minus five x is equal to zero. So this is going to happen when x is equal to zero. Let me not skip steps. So this is going to happen when x to the third is equal to zero, or four minus five x is equal to zero. For x to the third equaling zero, that's only gonna happen when x is equal to zero, and four minus five x equaling zero, we'll add five x to both sides, you get four is equal to five x, divide both sides by five, you get 4/5 is equal to x. So here these are the two places, where our derivative is equal to zero. Now are there any places where our derivative is undefined? Well our function right over here is just a straight up polynomial. Our derivative is another polynomial, it is defined for all real numbers. So these are our two critical points, or we could even say critical values. Now let's think of what g prime is doing on either side of these critical values, and I'll draw a little number line here to help us visualize this, and so. So there we go, little bit of a number line. Let's see we care about zero and we care about 4/5. So let's say this is negative 1, this is zero, this is one, and so we have one critical point at, let me do this in magenta, we have one critical point here at x equals zero, and then we have another critical point. I will do this at x equals 4/5. So 4/5 is right around there. So that is 4/5 and let's just think about what g prime is doing in these intervals, and these critical points are the only places where g prime might switch sides, switch signs. So let's first think about this, let me pick some colors I haven't used yet. So let's think about the interval from negative infinity to zero. So this is the open interval from negative infinity to zero, and we could just plug in a value, let's try negative one, negative one is pretty straight forward to evaluate. So let's see you have four, you're gonna have four times negative one to the third power. So that's gonna be four times negative one, minus five times negative one to the fourth power. So that's just gonna be one. So let's see this is going to be negative four minus five. Which is negative nine, so right over here, g prime is equal to negative nine, and so we know over this whole interval, since it's to the left of this critical point, we know that g prime of x is less than zero, and so our function itself is decreasing over this interval and so we know we need to go from increasing to decreasing, so you can already say well we can't go from increasing to decreasing at this critical point, because we're already decreasing to the left of it, but anyway let's just think about what's happening at the other intervals. So in the interval between zero and 4/5, so that interval right over there, so it's between zero and 4/5, well let's just sample a number there. Let's say the number, I don't know, 1/2? Might be fairly straightforward. So we can evaluate g prime of 1/2, g prime of 1/2, is equal to four times 1/2 to the third power. 1/2 to the third power is 1/8. So it's 4/8, or it's just 1/2 minus five times 1/2 to the fourth so that's 5/16 minus 5/16 and so this is equal to 8/16 minus 5/16 which is equal to 3/16, but the important thing is it's equal to a positive value. So in this blue interval right over there, and actually let me put 4/5 in a different color so we see that it's not part of that interval. So in this light blue interval right here between zero and 4/5, g prime, g prime of x is greater than zero. So we know our function is increasing, and so let's see what's happening to the right of this, and the easiest value to try out would just be one. So let's try out x equals one. It's in that interval. So when x equals one, I'll just write g prime of one is equal to four minus five. Four minus five, which is equal to negative one. So g prime of x is less than zero, g prime of x is less than zero. So our function, so we could say g is increasing here. It is decreasing, oh sorry let me be careful, g is decreasing here. The function itself is decreasing cause our derivative is negative. Then our function is increasing here cause our derivative is positive, and then our function is decreasing here. So at what critical point are we going from increasing to decreasing? Well we're doing that at x equals 4/5. So we have a relative maximum at x equals 4/5. 4/5, if they said, "Well where do we have "a relative minimum point?" Well that's going to be happen at x equals zero. We're going from decreasing to increasing, but we've answered their question of where do we find a relative maximum point.