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Problem 18 Medium Difficulty

Let $ f(x) = 2 - | 2x - 1 | $. Show that there is no value of $ c $ such that $ f(3) - f(0) = f'(c)(3 - 0) $. Why does this not contradict the Mean Value Theorem?

Answer

This does not contradict Mean Value theorem, because it is not applicable.
Hint : Look at the derivative of $f(x)$ at $x=\frac{1}{2}$

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Video Transcript

Okay. The question here is let every beck that goes to minus the absolute value to X minus one and show that there's no value of C just after three minutes after zero equals f prime of C times, three minutes. You and why does this not sounds for why this does not contradict you mean about him? Okay, so I think the best way to go about this particular problem is to think about the derivative and specifically around the sharp edge produced by this after by function. So this occurs when two X minus one is equal to zero. So when this is equal zero, well, this accused this equal terms when X is equal to one half. So remember, when you have a nap to value function is actually a piece wise divide function, and so you can you can, ah divide it up into two parts. So when X is greater than one happened when X is less than one half, so really affects can be re written as this. So if the function is less than one half, you put a negative sign in front of this function. So when if X is less than one half. Then it will be two minus. And then we put another negative Mix it positives of two plus two X minus one and then, if accepted, it's greater than one half. Yeah, you get the same exact function you get the function that's written. And then if you now you can take the derivative of dysfunction, boys Lee. So this will be a prime. A bag is equal to now. You just take the derivative of both of these function, which gives us two and negative too. So recall that these you're now you're getting two different answers at X equals one after this is occurring when exit less than one half. And this is occurring when X is greater than one. So you can obviously see here that the derivative at the point equals one half does not equal to each other. They're two different answers. And since our domain here is asking three to zero, I mean zero to three because we can see that with the way they set up the mean value, their arm equation right here on. We know that one half is in this interval. So since this is in the interval, you know that the derivative at one half does not exist. And remember the two conditions required for the mean value serum, our continuity and different ability and its domain in this interval, especially in the open interval. And since this condition right here has failed, we know that there is no way that the fact that this function this function affect failed the condition to fail the question fill the conditions required for the mean very theorem. We know for a fact that there's no value see such that affect three minus half of zero is equal. The F primacy times three minus zero and this is this's why it does not contradict them invalid because it doesn't even satisfy the conditions for