Problem

Use the guidelines of this section to sketch the …

12:41

Problem 6 Medium Difficulty

Use the guidelines of this section to sketch the curve.
$$y=x^{5}-5 x$$

Answer

$(0,0)$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

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Video Transcript

we are given the equation F of X is equal to X to the fifth minus five X, and we have to sketch this function. The first thing we should do is figure out the domain of dysfunction. And so this is just a normal polynomial. The domain will just be negative. Infinity to infinity or X can contain all real numbers. Uh, all real numbers and you can figure out the intercepts from here. So so what? The X intercepts first so we can set this equation equal to zero. You can fact throughout an X and we have X is equal to zero from this X over here. So have you figured out one x intercept? So what about this one right here? X to the fourth minus five is below zero. Well, we have We could move the five to the other side and take the fourth fruit of five. And we have not one but two. Ah, we have two routes here. So because it's to the fourth roots and four is an even number, so we can have to the the four through of five comma zero and the fourth root of, well, the negative for 3 to 5 zero. The four through 25 is around one point 1.495 So you can put that or you can have this. So what about the Why intercept all the white interests that to use plug zero in for X and they have zero minus zero. So the Y intercept is actually want to be 00 we can test for it's simply functions symmetry. You can do this by checking if it is even so if the original function is equal to if we replaced X with minus X. So we have minus X to the fifth minus five times minus. Now, we, uh, negative five negative x to the fifth plus five x. And since this is not the same thing as the original function, you can say that dysfunction is not even. So what about the odd test? Well, is whatever we got from our even function is that equal to if we negate the original function. So let's try to negate our original function. This is our original function, and we distribute this negative one a negative X to the fifth plus five. So they are actually the same function. So that means dysfunction is odd or symmetric around the origin. So it is odd there is no asking totes because this is just ah polynomial function. Uh, no asking, Toad. Since we've got all this information from the original function, we can differentiate the function and it would be five x to the fourth minus five so we can figure out increasing decreasing end any extreme, extremely just another word for minimum or maximum. So we can set our derivative Teoh zero we're left with X to the four is equal to one and X to the X is equal to the four through of one which is equal to plus or minus one. So we have two points of extreme. We have plus one, and we have minus one. Now we can plug in this X value back into the original equation. So for one, we have want to the fit that just one one minus five. That's negative. Four. And then we have negative one to the fifth. That's negative. One in the negative, one times negative. Five. That is, uh, plus five and then minus one plus five. That is four. So we can have a few intervals here Since we say that these X values plus one and minus one. This is when our slope of our attention line is zero. So we could have a few few intervals that we contest negative infinity to negative one and from negative 1 to 1 and then from one to infinity and then we can have a test number. So for this one, we can say our test number will be negative too. This test number could be zero and this test number, it could be too. And what is the sign of when we plug this number into our equation? So we have negative too. If we plug this but number into our derivative, we will get a We'll get a positive sign, plug zero into it. We get negative, it'll be a plug to into our derivative. We will get a positive sign so we can say that our function is increasing on negative infinity negative One union one to infinity and it is decreasing on increasing on negative 1 to 1. So let's move on to concave ity. To find concave ity, we have to find our second derivative. So the derivative of this. So that is 20 x cubed, So we can set this equal to zero. When we have an inflection of potential inflection. Point X is equal to zero, and we could test for concave ity by setting up two intervals, one that is less than zero. So negative, infinity to zero and from zero to infinity. Let's try to find a value that's in this interval. So I'll just pick negative one and one and then try to figure out is the sign of when we plug this value back into our second derivative. Negative. One cube is just negative one times 20. So that sign will be negative. Well, if we plug in one, the sign will be positive. So what is our con? Que Vitti? Well, for this one is con cave downwards. And for this intro, this interval it is going to be con cave upward. And our point of inflection will be X equals 20 So we have a point of inflection at 00 and we have con cave upwards from Oh, for on 02 infinity and then con cave downwards on negative infinity. Your so let's sketch what dysfunction would look like if you're familiar, familiar with with cubic functions, it looks something vaguely like that. So we could graft the X intercepts first. So 00 the We said that the fourth root of five is about born in the half. Well, see, that's I'll say that is one and negative one. So right here is our ex interested, and then we can point out, are extreme So negative one four roughly around here. No legal. This negative one comma four and then are other extreme Your other point? Ages one comma negative four And then this one We said it, Woz. Negative. Negative forth fruit of five. Comma zero. That is 00 And then this will be just the fourth fruit of five. Common zero. And then we are coming from the bottom were increasing cross the X axis. We go to this point That is our maximum, our local maximum. Then we go back down through the origin. You're decreasing from this point to this point, we're going to touch this point. That will be another. That will be a local maximum. Scuse me. Local minimum will come back up increasing crossing the X axis and then it will continue increasing. And then we have a point of conch avidity, right? A point of inflection right here at 00

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