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Fl) = 1 '} a* 3* H 9 9 that f([o; 1]) and show Athat fconunuous but that f is not monotone . at y 8 interval [ € 2...

Question

Fl) = 1 '} a* 3* H 9 9 that f([o; 1]) and show Athat fconunuous but that f is not monotone . at y 8 interval [ € 2

fl) = 1 '} a* 3* H 9 9 that f([o; 1]) and show Athat f conunuous but that f is not monotone . at y 8 interval [ € 2



Answers

If $ f $ is continuous and $ \displaystyle \int_{1}^3 f(x) dx = 8 $, show that $ f $ takes on the value 4 at least once on the interval $ [1, 3] $.

We are to show that this function satisfies. Really stare. Um and then find the C. Values that make it true. Since we are taking the cube root and then the and the other one is taking the two thirds power. We can fill any X. Value in because we can take the cube root of all positives and negatives. So first we will say that F of X is differentiable On the open interval zero 28. It's also continuous on the interval Closed in a rule 0-8. To back up the fact that it is differentiable. Let's look at the derivative. Our derivative Would be 2/3 Acts to the -1 3rd. Yeah -2/3 X. to the -2/3. We could rewrite that as to over three cube root X -2. over three cube root X squared. Which we could combine into a single denominator of three Cube root X squared. The first fraction we multiply top and bottom by the cube root of X. Yeah. And that's what we get now. That is non differentiable at zero. But that's okay. Just has to be differentiable on the open interval under the mean value theorem that derivative at sea which would be to cube root X minus two. Actually to cuba group C minus two over three cube root C squared Must equal zero at at least one point. Now for fractions equal to zero. It's because the top is zero. So let's set the top equal to zero. Let's add the two to both sides to cube root. See equals two Dividing side by two. So the Q. Bert of C Equals one. And then if IQ both sides, I get Sequels one that is in our interval. So that's where we're always theorem is satisfied.

All right. Were given that F is a continuous function and were given the value of an integral of F X. And were asked to show that ffx takes a particular value over a particular interval. The given information in the question, and what the question is asking for immediately suggests that we need to use the main value fear. Um specifically the mean value theorem for integral, excuse me, Which is a firm that states that the function F of X takes its average value at some point ah in the interval where the average value of the function F of X on the interval A to B is given by the formula on screen. Right now, that formula gives you the average value Ff. The mean value theorem for integral states that the function F of X takes the value F at some point in the interval A to B. So if it turns out that four is the average value of F of X on the interval from 1 to 3, then that will have solved this problem. So Let's go ahead and compute the average value of F on the interval from 1-3. So by the formula, we want to compute the average value which equals one over three minus one Times the integral from 1 to 3 of F of x D X. And this is the other reason why the phrasing of the problem heavily suggests that this is the correct way to solve this problem because we are not told what F of X is at all. So the integral from 1 to 3 of F of XDX is not something we can evaluate on our own. Fortunately, the problem tells us exactly what it is. A problem tells us that this integral equals eight. So we can compute this anyway, this expression equals one half times eight, which equals four. This is exactly the value. We were hoping that this is exactly what we were hoping the average value would be, and the result of the problem showing that F takes the value for that is exactly what the mean value theorem implies. So we're done.

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.

In this question, we have to find minimum value of ethics so I can write affects equal one minus two Upon XK Plus one. So effect will be minimum, Yeah, two upon Xk plus fun is maximum. That is Ex care plus one is please, so X care plus one is least when Then x equals zero When x equals zero. This minimum value of fx is f Judo Equal My Next one. So correct option is option is D.


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