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Problem 80 Hard Difficulty

A cubic function is a polynomial of degree $3$; that is, it has the form $ f(x) = ax^3 + bx^2 + cx + d$, where $ a \not= 0 $.
(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.
(b) How many local extreme values can a cubic function have?

Answer

(a) Proof in the video ;
(b) A cubic polynomial can either have two local extrema or no local extrema

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Video Transcript

a cubic function is a polynomial of decree three of the form F of x equal a X Q plus bx square plus C X plus D. Were a C from France era in part able to show that a cubic function can have 21 or no critical numbers. And we give examples and sketches to the straight. These was good barbie. We discuss about how many local extreme values can cubic function have. So if our a we're gonna start by remembering the definition of critical point, we know that see on the domain of the function is a critical point of F. Or a critical number. If the derivative at that point does not exist or it exists and is equal to Sue. So this is the definition of critical number we're going to use. And so first of all, we know that the the main of cubic function, is there real numbers? The set of real numbers? So if F is a cubic function, then the domain of F. Is the real numbers. That is the function is defined at every real number. That is because really not notice to find everywhere for any real number eggs, we have this and then the derivative of F is given by. We have to find the derivative of this general expression here, the three A. X square plus to be eggs plus C. And we know from the statement of the problem that A is not cereal. If A is not zero. Sorry. Here I made a mistake. Is three A. X square. Okay, Sorry. So because A it's not zero. This term is not zero and then it's a polynomial of degree two. But before saying that we can see that this derivative is defying everywhere. There is for any real number X. It means that this possibility is not it is not an option in this case. So you can say that f derivative is defined at avery real number eggs. And the conclusion of that is that the critical numbers of dysfunction, this cubic function are those real values X, where the conservatives equals zero. The only critical numbers are those values eggs in the real numbers such that the derivative of F X. You got to see that is because this option in the definition can happen in the case in the case of the cubic function, Because the derivative is also a polynomial in this case of Degree two and it's defined everywhere in the real numbers. So the only possibility for value X to be critical number of the cubic functions is that the derivative at that point is equal to zero. And it could be any principle can be any real number because it's the main of the cubic function is a set of real numbers. So we gotta find this equation here. But the equation I have to relative people to zero is the same as it's equivalent to to the equation of three. A X square plus to be X plus C equals zero. And we know that with a different from zero. This equation has to one or 0 boots then can have two, one or no roots. Okay. So if we take into account with the fit said before, it means that the function F can have two, one or no critical numbers. Hero, I put this way because we can have one. So in the case will be no route or non marketable number. So this is the situation we have a cubic function. Is there that cubic functions defined everywhere in the real numbers and conservative is also defined everywhere in the real numbers. And so the only possibilities for critical numbers for this function are the real values of X. For which the relative first robotic zero. And this becomes a second degree equation Which we now can have 21 on our roots. And then this implies that the function can have Only to one or no critical numbers. So let's put some examples of that. So I'm gonna take first the function X cubed minus three eggs square. That's two eggs. Which which can be factored out at six times X -1 Times X -2. Can be verified easily that this is a factory ization of dysfunction and the derivative three X squared minus six X plus two. Um has two different real roots. So in that case The function has two critical numbers. And the graph of these function with three rail routes X equals zero X equal to one X equal to. Is this one here? As we can see we have two values of X for which Derivative is zero about this point and about this point here. So this is an example of the function having to critical numbers. one of them corresponds to a relative maximum value here and the other corresponds to a relative minimum value. Another example is I got here that these Case of two critical numbers corresponds to to root for the first derivative. That is we find the first serve a tip of dysfunction three X squared minus six X plus two. We can see easily that that derivative has two different real roots and these two different real roots corresponds to curriculum numbers of F. So another example is X cube. So let me put here two pretty cool numbers now uh X cube. Which derivative is three X square. Which has only one route which is in fact the same route as sq Beauchamp's X equals zero. Yes, only one critical number zero. Let's see x equals zero. In this case if we draw the function X cuba get is graph here. And as the derivative represents this local detention line to the curb. You see that at zero we don't have a relative maximum maximum or minimum of the function because the function From one side to the other of zero has negative and positive values respectively. So it is not maximum or minimum. But the tangent line is just the X. Axis that zero. And so that is why this graph Which is always increasing as some kind of flat appearance near zero. Okay, so in fact the root zero mexican series, third order route because it's zero for the function itself zero for the first relative also a zero for secondary. So its third order There is zero of the function. But in in terms of relative extremists, not a relative extreme at all, but it is one critical point because there is only one route for the facility. And the other example we will show here is um if a vaccine equal X cubed plus X, which is the same as X times X squared plus one. The first derivative is three x square plus one. And this corresponds to no critical numbers. And that is because the expression here is always positive. X square is positive or zero, sensor is positive or zero. But when we had one is strictly positive so cannot be zero. So the first serve a tiv has no real zero. So the function has no critical numbers. And if we draw this function, we can see that. Have we had this situation because the first derivative at zero is 1. We don't have this flat appearance near zero as in the case of X cubed. But we have some kind of slope here which is just a slope one at the urgent. So they function don't have Real extreme extreme values at zero But it has not the flat appearance near zero. But again there is no extreme value and corresponds to not critical numbers. It's very important to notice that in cases two and three that is X cubed and execute Plus X. We have no extreme value. But in the case of execute we have one critical number which is the only route of the first serve a tive. In the case of excuse plus X, we have no critical number because the first derivative is not serious at any real number and that are the possibilities. And now we have already almost keep the answer to barbie. And that is we can only have to extreme values or no extreme violence at all. So a cubic function can have to extreme values. That corresponds to the first reality is having two different real roots. And in fact one of these critical uh numbers corresponds to a relative maximum and the other relative minimum. As in this case here about here. Mhm. And that's one possibility. But the other possibility is that he has no street value at all. And there is no other Possible way because in the case two and 3 that is one critical number, no vehicle number. We'll always have an inflection point. There is a change of conquering cavities or from one side to the other other point, christian value having different routes or it has no extreme values. I don't know until disarming possibilities.