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Problem 17 Medium Difficulty

Let $ f(x) = (x - 3)^{-2} $. Show that there is no value of $ c $ in $ (1, 4) $ such that $ f(4) - f(1) = f'(c)(4 - 1) $. Why does this not contradict the Mean Value Theorem?

Answer

\[
f(x)=(x-3)^{-2} \Rightarrow f^{\prime}(x)=-2(x-3)^{-3} \cdot f(4)-f(1)=f^{\prime}(c)(4-1) \Rightarrow \frac{1}{1^{2}}-\frac{1}{(-2)^{2}}=\frac{-2}{(c-3)^{3}} \cdot 3 \Rightarrow
\]
$\frac{3}{4}=\frac{-6}{(c-3)^{3}} \Rightarrow(c-3)^{3}=-8 \Rightarrow c-3=-2 \Rightarrow c=1,$ which is not in the open interval (1,4) . This does not
contradict the Mean Value Theorem since $f$ is not continuous at $x=3$

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Video Transcript

All right. So this question here is asking us to see let affect tickled ex ministry to the negative, too. And we're being told to show that there was no value of sea and one for shut step fo four minus after one equals f primacy time for my one. And why does does not contradict the Minbari therm. So before we jump into this problem, I think it's very important to mention that this statement and it just kind of seemed like they just threw. And right here is actually just a mean very idioms in a different form if you it's just rewritten in a different way. So if you bring the four minus one underneath, it would just be a promising because after four minutes, everyone over four minutes one which is simply the means that it's there. So, sensi we're going to evaluate find the sea. That would be a good way to for a start off this problem. So the first question be to find a derivative after back so primal backs. Well, this is a changeable problem. It would be best to apply to change Will. We're going to bring this negative two out to the front. And then we're going to subtract minus two minus one, which gives us negative three. And they would take the derivative of the inside. But Census X will be times one, so we don't write that. So this is simply are derivative here. And we are also going to evaluate our values of a foreign, everyone. So after four, just simply, he called too. Ah, one over one squared. Honey, this is not right. Oh, yes, it's one of the one squared. So this woods come out to be one if you just do the math for minus three is one. Yeah, and then after one comes out to be hey, completely one fourth. And you can also check that by flooding in that value. And then you'LL get one for with us. Well, and then So what we're going to do now is plug in these values into our mean very serum statement. So this will be one minus one fourth equal. And then this is saying after embassies, all we do is take a derivative and substitute to see intern where there's written X. So this will be negative, too. Time See? Minus three, too negative. Three time for minus one. Which is this tree? One minus one forthis, simply the way for it. Excuse me about this. All right about that, um And then negative two Time three is negative. Six. And recall that any exponents raised to the negative power can be written underneath, so as a fraction. So this will come down to the denominator. C minus three razed to the positive third power. Now and now we can rewrite this. We write this, you can multiply, see ministry on this side. And six times four divided by three. Take negative. Six times four, divided by three Gives us negative eight. This will be see minus three, razed to the third power Equal. Negative eight. And then now we simply take the cube root on both side and a Q boot of negative eight. Negative too on. Then we add three to both sides and that is equal to one. However, there is a slight problem with this. I don't know if you noticed, but the sea value. It's not empty domain. Not in the interval. Now try not to Indian trouble. It is not in one four. Because when it we were asked to evaluate a sea in one and the open interval from one before. So I mean who do not include the end point, which are X equals wanting extra before and we found a value of C equals one. So this is this value of C is actually not allowed in our in our interval. And there's actually a really good reason why this is not a lot. Um, the reason why is because if you look at the original function after sex, it closed was X minus three to the negative two. Let me make sure it's right. Yes, it is. Remember, this could be re written as one over X minus three square and you could if you notice now that there's actually this continuity in the domain of this function and it occurs when the bottom equal zero and the bottom echo zero when X is equal to three. Because if you plug in to be right here, you get three ministries dio and one over zero squared is one of zero and one of reserve is undefined, and we are in the interval from one to four and three is included and wanted for so there is actually a dis continuity in our function and the in the interval and the open injury will want to four. So that means that the conditions required for the mean value zero continuity is failed. And since this is failed, you can't defense ability at three Mexico story also does not exist. So now you have you have both conditions of of the mean value system failing with this function. And that explains why this value of C equals one does not fit in order in our interval. What and that explains why they're just not contradict the mean value to him because it is not. It doesn't even satisfy the condition for the memory thing. So that's why it's really important to check for the whether the function even passes the conditions first before even solved the problem.