Ολικά ελάχιστα και μέγιστα (ολόκληρο το πεδίο ορισμού)

- [Voiceover] So we have the function g of x is equal to x squared times the natural log of x. And what I wanna do in this video is see if you can figure out the absolute extrema for g of x. So are there x values where g takes on an absolute maximum value, or an absolute minimum value. Sometimes you might call them a global maximum, or a global minimum. So the first thing I like to think about is well, what's the domain for which g is actually defined? And we know that in the natural log of x the input into natural log, it has to be greater than zero. So the domain, the domain is all real numbers greater than zero. So x has to be greater than zero. Anything lateral log of zero is not defined, there is no power that you could take e to to get to zero, and natural log of negative numbers is not defined. So that is the domain. The domain is all real numbers such that all real numbers access so that x is greater than zero. So our absolute extrema have to be within that domain. So to find these, let's see if we can find some local extrema and see if any of them are good candidates for absolute extrema. And we could find our local extrema by looking at critical points, or critical values. So let's take the derivative of g. So g prime, using a new color just for kicks. All right, so g prime of x is equal to, we could use the product rule here. So derivative of x squared which is two x times the natural log of x plus x squared times the derivative of natural log of x, so that is one over x, and I can just rewrite that. X squared times one over x. And we're gonna assume that x is positive. So that is going to be, that is going to be just, and actually I didn't even have to make that assumption for what I'm about to do, x squared divided by x is just going to be x. All right. And so that is g prime. So now let's think about the critical points. Critical points are where the derivative they're points in the domain, so they're gonna have to satisfy x is greater than zero, such that g prime is either undefined or it is equal to zero. So let's first think about when g prime is equal to zero. So let's set it equal to zero. Two x natural log of x plus x is equal to zero. Well we can subtract, we could subtract x from both sides of that and so we get two x natural log of x is equal to negative x. See, if we divide both sides by x and we can do that, we know x isn't gonna be zero, our domain is x is greater than zero. So this is going to be, actually let's divide both sides by two x. So that we get the natural log of x is equal to negative one half. Negative x divided by two x is negative one half. Or we could say that x is equal to e to the negative one half is equal to x. Remember, natural log is just log base e. So e to the negative one half, which we could also write like that, e to the negative one half, or one over the square root of e. So that's a point at which g, at which our derivative I should say is equal to zero, it is a critical point or critical value for our original function g. So and that's the only place where g prime is equal to zero. Are there any other points where g prime is undefined? And they're have to be points within the domain. So let's see, what would make this undefined? The two x and the x, that you can evaluate for any x. Natural log of x, once again, is only going to be defined for x greater than zero. But that's, we've already restricted ourselves to that domain, so within the domain any point in the domain our derivative is actually going to be defined. So given that let's see what's happening on either side of this critical point. On either side of this critical point. And I could draw a little number line here to really help us visualize this. So, if this is negative one, this is zero, this is let's see, e to the, this is gonna be like one over, oh boy this is, this is going to be a little bit less than one, so let's see, why don't we plot one here, and then two here, and so we have a critical point at one over the square root of e, and we'll put it right over there. One over the square root of e. And we know that we're only defined from, for all x's greater than zero. So let's think about the interval between zero and this critical point. Right over here. So the open interval, from zero to one over the square root of e, let's think about whether g prime is positive or negative there. And then let's think about it for x greater than one over the square root of e. So that's the interval from one over square root of e to infinity. So over that yellow interval, we let's just try out a value that is in there. So let's just try g prime of, I don't know, let's try g prime of 0.1. G prime of 0.1 is definitely going to be in this interval. And so it's going to be equal to two point, two times 0.1 is equal to, is equal to 0.2 times the natural log of 0.1 plus 0.1. And let's see. This right over here, this is going to be a negative value, in fact it's going to be quite, it's definitely going to be greater than negative one. Cuz e to the negative one only gets you to, let's see, e to the negative one is one over e so that's one over 2.7, so one over 2.7 is going to be, so this is going to be around 0.3 or 0.4. So in order to get point one, so this'll be around 0.3 to 0.4. So in order to get to 0.1 you have to be even more negative. So this is going to be, I could say less than negative one. So if this is less than negative one, and I'm multiplying it times 0.2, I'm gonna get a negative value that is less than, less than negative 0.2, and if I'm adding 0.1 to it, well, I'm still going to get a negative value. So for this yellow interval, g prime of x is less than zero. And it would be, I should have gotten a calculator, or I could have gotten a calculator out, I could have just evaluated a lot easier. So g prime of x is less than zero in this interval. Now let's see in this blue interval what's going on. And this'll be easier, we could just try out the value one. So g prime of one is equal to two times the natural log of one plus one. Natural log of one is just zero. So all of this just simplifies to one. So over this blue interval, I sampled a point there, g prime of x is greater than zero. So it looks like our function is decreasing from zero to one over square root of e, and then we increase after that. And we increase for all x's that are greater than one over the square root of e. And so our function is going to hit, if we're decreasing into that and then increasing after that, we're hitting a global minimum point, or a absolute minimum point at x equals one over the square root of e. So let me write this down. We hit a, we hit a absolute minimum at x equals one over the square root of e, and there is no absolute maximum. As we get above one over the square root of e we are just going to think about what's going to be happening here. We're just going to one, we know that our function just keeps on increasing and increasing and increasing forever. And you could look at even this, x squared is just gonna get unbounded towards infinity, and natural log of x is gonna grow slower than x squared, but it's still gonna go unbounded towards infinity. So there's no global, or no absolute maximum. No absolute maximum point. And now let's look at the graph of this to feel good about what we just did analytically, without looking at it graphically. And I looked at it ahead of time. So let me copy and paste it. And so this is the graph of our function. So as can see when this point right over here, this is when, this is one over the square root of e, it's not obvious from looking at it that it's that point. X equals one over the square root of e. And we can see that it is indeed an absolute minimum point here and there is no absolute maximum point. There's arbitrarily high values that our function can take on.