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Problem 57 Hard Difficulty

(a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.

$ \cos x = x^3 $

Answer

(a) $f(x)=\cos x-x^{3}$ is continuous on the interval $[0,1], f(0)=1>0,$ and $f(1)=\cos 1-1 \approx-0.46<0 .$ since
$1>0>-0.46,$ there is a number $c$ in (0,1) such that $f(c)=0$ by the Intermediate Value Theorem. Thus, there is a root
of the equation $\cos x-x^{3}=0,$ or $\cos x=x^{3},$ in the interval (0,1)
(b) $f(0.86) \approx 0.016>0$ and $f(0.87) \approx-0.014<0,$ so there is a root between 0.86 and $0.87,$ that is, in the interval
(0.86,0.87)

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Video Transcript

you park A. We're going to prove that the equation X cubed equals co sign of X has at least one real root barbie. We assure the calculator to find an interval of length 0.01 that contains a route. So we are going to define for that the function F equal X cube minus co sign of X. And we are going to consider that function. So first about before saying that to function less say the following. We know from this we want to final a root of this equation. That is Value Acts for which ethical zero. Mhm. So you say that If the function is secret zero we can say that X cube is equal to co sign of accessories. A root of this function is a solution to the given equation. But in fact what is important here is to notice that this implies that the absolute value of X cubed is equal to the absolute value of cosine of X. And then the answer value cube of X. Okay the sequence of the absolute value of coastline of eggs and we know that the delivery of coastline is less than or equal to one. This will implies that the absolute value of X cube is less than or equal to one. And from these taking spirit of size we know that this implies that the absolute value of X is less than or equal to one. That is X belongs to the interval negative 11 close both and points. And that means that the solutions to this equation here can only be in this interval. So we we restrict our attention and this interval. So that's why we say that. Uh huh. That the function F is defined inbred interval. Okay. Because there only there we can find a solution of the aggression. That's one thing. The other thing we can say before given a solution to the problem is that if we sketch the graph of these functions that is the function X cubed and co sign effects, you know, excuse is something like this. Good and co sign is something like this. And here we can have some kind of doubt about the behavior but there's no doubt really because we can say something here as you can see the function Co sign is positive between zero and the value by half here and is decreasing between Syrian by half. The function x cube is increasing is positive and increasing between zero and plus infinity. It means they cut in one point. They intersect in one point at one point here let's say this point here and it seems to be the only positive solution to that equation that point here. But from the left side we can say something here. Is that the function X cube is negative while the co sign is positive up to negative by half. So if we look at the value of the function X cuba by half then we can say that the graph will be way below the co sign of faith and that's the case because if you calculate um that is by health cube, I have cube predicting here. He's about -38 75 3.875. And means that at this point where cause and begins to be negative The function excuse has a value of negative 3.8 because go sign can be uh Up to the value -1. That cannot be lower than that. It's impossible for the to your curves to intersect on the negative values of X. So there is no solution in danger of -10. So we know Moreover that the solution is only interval 01. So we can say this here uh four X belonging to a natty 10 which is a negative part of the interval. We know there lies the solution for X. In that interval we know that co sign of X is positive because that's another way of saying different from what I see here that is, we know that -1 which is uh the minimum value that can attain the solution of the equation at that point. Co sign is a positive value has a positive value. That's because the co sign, I'd see it again here if you want here we have negative plus by half and negative by half about 1.57 That is -1 is like this and there we have co sign with the positive value. So being cause I'm positive on this interval and X cube negative and that interval mhm. These two things together imply that X cubed equals co sign of eggs has no solution on zero negative 10 mm. Mhm Mhm. And then knowing that the solution can only be on the interval negative 11. And that there is no solution on the interval negative 10, then the equation X cubed equals co sign of eggs can have solution only in sierra one. The solution or solutions if there is more than one lies there. Okay so all this we have said without calculating anything that is we are only analyzing the functions through the definitions of the function in this case using the properties of coz i the actual values less than or equal to one. And here is in the science of the functions on the intervals With only that we can we have constrained our attention to the interval 01 now because if this continues. So f from 01 to the real numbers, the finest X cubed minus cosine of X is continuous. What we need to do is to find an interval where the function changes sign but the interval will be the same interval here. We need a close intervals of course and this close interval and let's see that the function changes sign there. F at zero is equal to zero cubed minus go sign of zero, that is 0 -1 that is -1 which is negative. an effort. One is one cubed minus co sign of one. That is one minus co sign of one. And co sign of one. We know is positive number but less than one. If we see the graph here of co sign here here is the the maximum value of course in which is one. The interval native by health by half and uh mhm we can say that there you hear is probably half and we see clearly that At the value one which is decide here, which is less than by half. We know that the co sign of that one is positive is positive and it's less than one because Co sign is only one at 0. So this number here is a positive number Being close in one. Less than one. So the function changes sign at the end points of the club of the close interval and the function is continuous. We know from that that the functions can have zero from the internal zero water. And uh moreover, the route is inside the intervals not the endpoints because at the end points The functions in the zero. So let's write it here. So F is continuous. Only closed interval. Yeah. And changes sign on the endpoints of these interval. More precisely At zero. The functions negative and a one The function is bust. Then there exist at least a value eggs on the open interval such that F of x equals zero. That is x cubed minus cosine of X equals zero. That is exc your equal co sign of exit is there is at least one solution of the equation which we wanted to prove. So that the main result. Yeah. So we have proved the first part. That is there is at least one real wood for the equation, the human equation. And we know more over that. The solution is on the Internet, In fact, if we draw, if we sketch the graphs of the functions together, we can see that the solution is uh huh, Closer to one than 0 mm But now let's calculate that. We got to find an interval of length mhm And an interval of length 0.01 that contains a route to the equation. So for that we do kind of by section method which consists in the following we can we start with the interval 01 cutting in house by the mid 0 .1. And we know that we can apply the same theorem here for these two interval studies, we evaluate the function at the end points in this case. We have evaluated already at zero already at one. We know that at zero remember is negative and that one is positive And now we evaluated zero points uh 0.5. We do that you get yeah -0.75 or less. So at 0.1 the function is negative and so the change of sign now happened on the interval 0.5 one. Yeah. So applying the same theory, we know that there is a route Now in the interval 0.51 and we do the same again. We could the interval zero point I want in the half And that's a midpoint, there is 0.75. And you always look at the interval where the sign changes and going for this way and this call by section method and with that we do that until the interval which contains the solution is office size we want that is the precision is given by the length of the interval containing the world. In this case if we do this procedure uh sufficiently large or for many durations we can find yeah very good estimation of the route. And in this case we we can find the following value. So I root for the equation equals zero is zero point 86 54 74 033 10 16 13. Here we have used a very good precision that is. We stop the process when the interval world Containing the route was about the length of 10-2014. So it's very good approximation. And knowing that the route is about that value, we can say that an interval containing the route And of land, 0.01 is the following. So the interval 0.86, contains the route because these values a little bit greater than this and less than this and The length of this interval is 0.01. We always can do that. That is we do by section procedure or method up to a very good precision. And then we find an interval with give a measure that contains the route. We can give even and even smaller than this interval that contains the room because we have here in fact Opposition of 10 to the -14 About that. So we can do we can give intervals of of smaller size and this containing Taru. And as a final remark, we can say that this is the only road of this equation On the Interval 01. We can verify that for example, graphically if we sketch the graph of X cubed and co sign of eggs over 01. And we can verify that is the only world