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University of North Texas

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Problem 21 Medium Difficulty

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \dfrac{1 - e^{1/x}}{1 + e^{1/x}} $

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Video Transcript

we want to use cast software to grab of X is equal to one minus. I need to the one of ex all over one plus e to the war breaks Then use that same software to find the person So I could derivatives and grab those derivatives Estimate intervals were the punctures increasing decreasing intervals of calm cavity inflection points at any extreme values that the function might have. So I went ahead and graft ffx here already. And I chose this window here cause I just kind of zoomed out a little bit. And then I saw as ex went to infinity Negative family. It looked like it just approached Y z zero and there wasn't really any other important information I thought. Then I went ahead and already graft and found the first derivative. And grab that over here on the right and before we actually find our intervals where the function is increasing, decreasing, notice that on our original function at X is equal to zero. The function is undefined since we have this kind of piece wise function going on right here. So now what we need to go ahead and do is keep that in mind when we are finding where the functions increasing and decreasing Because we know even though it looks like our first derivative should be defined at X equals zero, we should know that we shouldn't get a value for it. So our interval shin always exclude zero in this case. All right, So remember, we know a function is increasing when f prime of X is struck a larger than zero. And so starting from the left, it looks like from negative infinity up to zero. It's positive. And then 02 and 30 and it will be decreasing. Went F prime of X is less than zero. And from this graph, it doesn't really ever look like it will be, um, less than zero. So there are no intervals of where the function is decreasing. But even just looking at EPA, Becks, it does look like it is always increasing. So from this, we can conclude that we have no local men's or local. Max is so no local man. Flash Max. All right, now let's go ahead. I go to the second derivative here, and all I did to get this window was I just zoomed out far enough to ensure I got all my ex intercepts. And those X intercepts occurred at negative 0.4170 point 417 And again, even though it looks like X is equal, zero would be one We need to go ahead and exclude X is equal to zero from this. All right, Now, if we want this to be Khan caged up, we want to find we're after will cry mystically larger than zero. So that's going to be from negative infinity up until negative 0.417 union with. Then start, I get zero up into 0.417 So that will be our interval where it is calm. Keep up now the function will be calm, caved down when f double prime of X is strictly less than zero and that would just be the rest of our interval. So negative 0.417 to 0 Union 0.4172 So now our two x intersects, including excluding excessive observe Well, to the left of this, it is Kong k up to the right of it. It's calm kicked out. So that tells us lab that inflection point here, and similarly, or are other Exeter's you'll be increasing to the right decreases conch, A book to the left, the conclave down to the right. So this is also a point of inflection. So we found our points and inflection. We found where the functions con que about cranking down we determine the function should always be increasing. And we have no local maximums for events, okay?