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3 points' Outline proof;, beginning with basic properties of the real numbers of the following theorem if f [a; b] 0,0) is continuous function such that f (r) ...

Question

3 points' Outline proof;, beginning with basic properties of the real numbers of the following theorem if f [a; b] 0,0) is continuous function such that f (r) for all x 6) , then f(a) = f(b)

3 points' Outline proof;, beginning with basic properties of the real numbers of the following theorem if f [a; b] 0,0) is continuous function such that f (r) for all x 6) , then f(a) = f(b)



Answers

Proof Prove that if $f$ is continuous and has no zeros on $[a, b],$ then either $f(x)>0$ for all $x$ in $[a, b]$ or $f(x)<0$ for all $x$ in $[a, b]$

So in the cushion the fountain of what it is doing there at square 1 to 6 cents per se. And the internal value of is one and being used to be. And you have the tools. The function is continuous for one body. Works in Colombia. In the in development from A. Three. The function it's convenient that affects the problem polynomial function of the body. So so you know they're calling the convenience for all value effects just or goodness more or less. Mhm. We will check the duty at one hand three. So then exit village keep close to one. So the value of the function F. One people 11 is six plus E. 116 Let's say that in the past three. And the limit accidents human bliss. But that's that is X. Four months six explosive. It is also cost a three man. It could be close to 45 So just convenience. And the point of the customer have exposed to three. There for three will be questioned. This government is syncing between, that's where that costume man, month 13 plus eight. So then we'll be custom one is one. And the limit and let me let you limit at as opposed to three. It was called minus six explicit. That will be also minus one that we conserve three. So we can see that the continent's company has 43. So from this condition we can say that that something is confident they double for instance excess carbon six cents. This is continuous giving us and then never one. Okay so and we have to prove that that must have at least 10 in the interval echo movie. And you can you can see there fo but the question everyone is three from 1 to 3 and for three years one is one. The video that is greater than zero and therefore three agent one is one that is less than zero. Okay. Yes. So and it is continuous with the electable van from a. T. So the graph of this function will use something like this. Do you the drop of this constant give you something like this. Okay. This point is three, this is school where it is or and I for one is for state if our money is positive support this is open and for three years indicating so you can see them from the draft. There is a Mhm. At the age of one point once you're in the lobby. So from the beginning and the three there's a 10 that is to at the age of the functional zero. So from the draft you can see them everybody. Is it The value of F. One is for state and the value of every place negative. So function eight weighing from posted by luna get by. So this function will cross that excessive. So there are there is at least 10 Its inventor then they suspension. So some drugs. Yeah. I guess you did efforts. Is company is concerned and the police inevitable a former D. And this sometimes complains advances continued and then tomorrow the clothes and a car. Okay? So you see that so hence book. I hope you understood. Thank you.

Function half of x equal to two times x. q negative three times x square negative 36 times x positive 14. And acts equal to see you equal to one. So here, in part a we know that We have to show that the function Africa continues for all values of acts in the interval 8 to be. So here You can see that function F is a polynomial of degree three. So we know that volley normal function is continuous at every value of acts. So F is a continuous for all values of acts in the interval cereal to one. So here in part a function have is a polynomial of 33. So function F is continues for all values of X in the interval Closing to 0- one. So here all finances for part a function F is a polynomial oh, degree three. It is our final answer for part A. Now with soul, part B. So here we have to prove that F Most have at least 10 in the interval 8 to be. By showing that F. Of A and F. Of we have a positive sign. So first of all we have to find the well of F. Of A. And F. B. So here let we checked whether F. Of seal and F. Of one how a positive sign. So here you can see that Fosco equal to two times 0 to the power three, negative three times zero to the power to -36 times you Plus 14. Now we simplify this and we get 14 and here We have to find a 4th 1. You call to Two times going to the Power three. Now I do three times went to the bar to negative 36 times one Plus 14 equal to here we get negative three. Sorry, 23. So here you can see that there is 10 an interval 0-1. So here we can say therefore there is at least once you in the interval 0 to 1. So here you can see that F of zero equal to 14 and 1/4 1 equal to negative 23. So it is a final answer for part B.

All right, so, we're given a lot of information and um we're told that F of A and F of B have opposite signs, so F of A and F of B have opposite signs. Then using the mean value theorem, or rather the intermediate value theorem, then somewhere between A and B. If they have opposite signs, so one of them might be down here and one of them might be up here, Well, they must reach zero somewhere in between. So, according to the intermediate value theorem, it must reach zero somewhere in between. Now, let's look at Raleigh's theory rallies theorem says that if there are two points that are the same. So let's say, for example, I'm going to call this C one and then I'm going to call C to another point where the function is zero. So, if if, according to Raleigh's theorem, F of C one equals F of C two, and I can arbitrarily state that C one, f f C one is zero. Because we know from the intermediate value theorem that uh somewhere it's going to be zero, so I'm just going to call that that X location C one. So if they're the same then yeah. Um The slope is zero somewhere. Yeah, but that's a contradiction because we were already told, and I didn't write it down that the slope is not zero. So we would therefore prove by a contradiction that F one of seed does not equal F two. F that F F C one does not equal F of C two. And so after the FFC two cannot be zero, it has one and only one solution between A and B. Only one and only one place where the function is zero.

Here. We want to show the function. Show the function is continuous for all values of X in the interval. And prove that f must have at least 10 in the interval. So here we're gonna be given aftereffects is equal to the text cube, my history expert. Mhm. Um -36 x plus 14. Mhm. What we see is that this is a polynomial so we know immediately that it is going to be continuous. There's no restrictions to it. So it's a continuous polynomial of degree three. Um And then we want to um Look show that it has at least 10. So we're going to use the intermediate value theorem here. Um Or Yeah. And then we know that a zero. So we're going to look between zero and one. Well, we know that F of zero. It's 14 And F of one is -23. So between zero and one we have this gap. Then we know that at some point it's a continuous function. It has to go through um zero point.


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