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