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If f(x)=x+1 f(x) satisfies the hypothesis of Mean value theorem on the interval [-2,4]Find a number € in the open interval ( _ 2,4) that satisfies the conculs...

Question

If f(x)=x+1 f(x) satisfies the hypothesis of Mean value theorem on the interval [-2,4]Find a number € in the open interval ( _ 2,4) that satisfies the conculsion of the theorem

If f(x)=x+1 f(x) satisfies the hypothesis of Mean value theorem on the interval [-2,4]Find a number € in the open interval ( _ 2,4) that satisfies the conculsion of the theorem



Answers

Determine whether the given function satisfies the hypotheses of the Mean Value Theorem on the indicated interval. If so, find all values of $c$ that satisfy the conclusion of the theorem. $$ f(x)=-x^{2}+8 x-6 ; \quad[2,3] $$

All you need for the main value theorem is a function needs to be differential and continuous. So in this problem because it is a polynomial, it is differential and therefore continuous everywhere. Ah If you don't believe me, graph X squared and you'll see that you have a U. Shaped parabola. Um And its differential and continuous everywhere. Especially from -1-7. Which is what we care about. Um Okay so the whole thought of this is that the derivative of this function which is equal to two X. Needs to be equal to Um f. of seven. Which when you square seven you'll get 49 minus negative one when you square it because your possible on over seven minus negative one. So when you do that arithmetic and I'm gonna change this X. Value to be a C. Value. Um Just some books do it that way. Uh If you think about 48 divided by eight. Uh the arithmetic care you'll get six. If you divide both sides by two you'll get the C. Value is equal to three. Uh I guess that's it.

First. Let's verify that we can use the mean values here. Um So we're gonna grab half of X equals E. To the negative two X. And then what we end up getting is Looking from 0 to 3. We see this is going to be continuous and differentiable. So we can consider F of three -F. of zero mm divided by three minus era. So you see the slope that we're looking for is a negative 0.33. And we're gonna solve for where this is the case. So we'll look at F prime of X. Which we know is going to be a negative who E. To the negative, correct. It's important to note that F. F. Three is going to be E. To the negative six -1/3. That's where this is coming from. So we want to determine what value of acts um will allow us to get that same slope and we end up getting, it's gonna be negative one a half times the natural log 1- is the 6th over 6th. And that's gonna be our final answer.

So here I have recorded the mean value theorem for us to refer to if we have a continuous function on the closed interval A to B. And its differential a differential between A and B. The average rate of change from A to B is the same as the instantaneous rate of change the derivative at some see on the interval. So first we just need to show the mean value theorem applies. So is f continuous on this interval? Well, yes. Right. Because it's a quadratic that's well known to be a continuous function is F differentiable On the open interval negative 4-6. Well, yeah, the derivative of a quadratic linear and a linear function makes perfect sense that any x value. So all that stuff seems fine. Let's now find the average rate of change. So we need F evaluated it negative for an F. Evaluated at six. So if we plug in negative four, we get 16 -4 or 12, and f evaluated six is 36 plus six or 42. So the average rate of change is the difference in wives over the difference in X values. We end up with 30/10 or three. So we know the target slope that Rafter is three. Let's find the derivative. Now we get two X plus one, so that means to C plus one needs to be three. In solving for C. We get one and note one is definitely on the interval negative 4 to 6. So that's our value of C guaranteed by the mean value through

Alright with the main value through all, you need to double check is that your function is differentiable and continuous? Um And because it's a polynomial, we can check that off because all polynomial czar differential and continuous everywhere. So as we look at this function defined as X to the fourth minus two X squared. And what we need to do is figure out what F of negative three is Because if you take the to the fourth power becomes a positive three times three times three times three is 81 and then uh three square it is um -3 When you square it's nine uh times think of two is 1918. But instead of actually figuring out what that value is, Um you also need to figure out whatever positive three years And I think it's obvious that you're going to get the same thing that three times three times three times three is also positive 81 and three temperature is also nine times two is also 18. So these two things are equal. And because of that when you use roll steer um Well this is the mean value there, but this actually rolls them too because the y coordinates are the same. Um That when you take the derivative of the function you want to equal to the slope. Well, since the y values are the same, the slope must be zero. Even though the change in excess is 6, 0 divided by six is still zero. Uh so what do you need to do from here? Is factor out the four X. You're left with X Square -1. Only if you remember, we only want the values between negative three and positive three. But when you go to solve for this, I'm going to change these access to be seized because they asked them as c values seeking equal zero or c squared minus one could equal zero. When you add one over in square root, C could be positive and negative one. And all three of these answers are between negative three and three, so we need all three of them.


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