WEBVTT
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the upper right hand corner of a piece of paper
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. 12 inches by eight inches. As in the
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figure is folded over to the bottom edge. How
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will you fold it so as to minimize the length
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of the fold. In other words, how would
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you choose X. To minimize why? And this
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is the statement we're going to stick with. This
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one here is clearer than the other one that is
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. We want to choose X in order to minimize
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value of Y. And for that we need to
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write why as a function of X. And they
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apply and then apply calculus to minimize the dysfunction.
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So this is the situation geometrically. And as we
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can see the fall the a sheet of paper and
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we found a right for angle here which is the
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reflection following the lion of length Y. Of the
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right triangle with the dot lines. So these two
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right triangles are the same this with the red lines
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and the other with this red line as his thought
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lines here because we have a right angle here of
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course. And that right triangles. Just this one
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here. So the triangles are the same. We
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have these length, we are going to go Z
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. That is the length of the start line here
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. The other dot line is eggs. And this
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diagonal red red line is why? Yeah. So
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we know that here, we have the total length
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minus x. That is eight minus x. Mhm
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. If we draw perpendicular lion here from this point
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here to the other side of the sheet of paper
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that is. We draw a vertical line. We
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have then um another triangle here and this. I'm
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going to call this distance from here to here.
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That is to this note, we're going to call
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it in And these other two here from this note
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here to the age of the paper, we're going
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to call it M. And from the figure is
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clear that M plus M is Z. Okay,
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this is something that we can see clearly clearly in
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this figure. Okay, so we want to write
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one in terms of X. And for that we
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are going to use these right triangles we have thrown
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here. We have the first one is deals with
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the red lines. The other one is this here
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using their bottom right corner of the paper. Mhm
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. And from the observation that is rectangle here up
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is the same as this rectangle with red lines.
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We can see that this side right here is of
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length X. Because it's the same as this when
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we fall the paper. Is this just the same
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? And the other side is this one is this
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one? That is this is see here, mm
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haven't done this when we now have all the red
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triangles. I'm going to draw them here. Mhm
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. The first one is the the single and uh
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red triangle. This one here. Yeah, with
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the right triangle here, the right angle here and
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X. Z. And y. The other one
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that's called these one what? Then we have the
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other is this here. So this is like this
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, this is the right angle and we have eight
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minus X. We have X. Here as a
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hyper luminous. And the other side is m You're
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going to call it too. And this one here
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, yes. Just like this with the hype oddness
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. See here is the right angle and here is
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we have called it in and we know that this
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side here is eight because he's the with of the
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piece of paper and we are going to cold is
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three. So we have now the three red triangles
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we're going to use and this statement here. So
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first yeah We use this one from yeah from right
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triangle one. We have the first relationship is X
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square plus Z squared equal y square. Okay remember
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the hype redness of this triangle here is why that
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is why the equation is like this by to ground
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zero. So we have this one. Mm So
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right I the first one, the second thing we're
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going to write is this Relationship here from these triangles
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of from two. Okay, mhm Eight minus X
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square plus m square. But like square. Yeah
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. And from the right triangle. Uh this one
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here the third one. Yeah. And so we
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have eight square plus and square equals Z square.
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Yeah. Okay, so we have this and we
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call this equation here two. Okay And this equation
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here three roman meditation. So these are three equations
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from the three right triangles we constructed from the figure
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. And so we start with now we recall that
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we want to write wine terms of X. And
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from the first red triangle we have Y here in
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terms of X and Z. So we've got to
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put Z in terms of X. And we will
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be done Yeah. And we'll be done for the
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first part of course. And then we have Z
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to put me in terms of X. We have
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this third equation you see is written in terms of
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end but we know em in terms of X from
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two and we know that N. M. And
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Z are related but this is rich. That's gonna
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give us what we want. So now we start
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with the second equation too. Mhm. Okay so
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we get from this equation here that eight square minus
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16 X plus x square equals plus m square equals
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X square. Ex queries cancel out. So we
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get from here that uh m square Plus 64 is
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equal to 16 eggs. Yeah you have the situation
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here and we can get a little bit further,
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we get that X. Is equal to m square
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plus 64 over 16. Or sorry maybe it's not
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accept. We want Sorry sam. So we add
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the other step we can do is M square Equals
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16 x-64. And then mm can be written
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as the square root of 16. My 16 X
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-64. We take a con factor 16. Here
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we get x minus four. And so this is
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four squared of x minus four. And in order
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to do this, We get to impose the condition
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that X-4 is positive or zero. So X
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minus four squared. And then record zero that his
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eggs Is at least four. That's very important condition
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. That is we cannot do this. Uh Less
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than four. If you do this, less than
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four we will we won't have a geometrically possible figure
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. Okay, so we have these we have these
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partial results were going to use after and then we
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have written mm in terms of X already and these
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were going to call four. Okay, Now use
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the other equation here is three. Yeah. From
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three. We get that. Uh eight that 64
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plus N square equals E square. But we know
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that N plus M equals E. And that is
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the observation we do right here following the construction and
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plus M. Give us all this whole land sea
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here so we can food and as Z minus M
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and 62. That here. So. And then
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64 plus Z minus m square is c square.
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That is 64 plus C square minus two. Zm
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plus m square equals the square. See squares cancel
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out. So we finally get to Z M equal
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64. Blossom Square. And that is why we
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are going to use this preliminary result here. M
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square plus 64 60 next. So we can as
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we can see here we have 64 M squared plus
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64. So we can replace that with 16 eggs
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. Yeah That means that two ZM is equal 16
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eggs. And for that Z is 16 X over
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two M. Which is eight X. Over M
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. And we get to put em. Now we
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found here mm you've seen four. Z is equal
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to eight x over for Square. It affects-4
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. That is Z is eight over four is too
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. So we get to in the numerator had two
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eggs over spirit affects one's four. And that's it
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. So we have see in terms of X and
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we are done too right? Why? In terms
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of X. So This is question one. So
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from one and five, That is this one here
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we obtain that? The C. X square plus
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the square. That is X square plus two eggs
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Over square. It affects-4 square equals Z square
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. A white square, sir. Mhm. And
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then why square is X square plus four X square
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over square. Sorry, over x minus four.
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Sure that is X squared. Common factor of one
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. Plus for over x minus four. That is
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six square times X-4 Plus four. Over X
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-4. These four canceled here. And so we
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get X square Time Sex Over X-4. And
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so why square is equal to X cube over x
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minus four because all the lens are positive. This
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we can take root and get y equals square the
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facts cube over X-4. And for that s
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get it, get it x gonna be positive and
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greater than for in fact very left for would be
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the same as positive. And I put it right
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here. So this is our equation of Y.
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In terms of X. And the condition will be
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excreted for And now for these two have sense,
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X cannot be four. In fact can be must
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be creators strictly than four. And from this forget
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that it is positive and the negative excuses. Makes
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sense. So this is our equation. Now we
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got to apply calculus here to find the maximum.
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Well the minimum value of Y and the corresponding X
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. So for that we need the derivative. So
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the function we are going to use this square root
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of execute Over X-4. And we need to
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calculate the derivative of this function. Let's write it
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as X cubed over x-4 to the one half
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. And with that mhm. Here, I'm going
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to write that. We're going to minimize f of
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X Is the goal. So the derivative is 1/2
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times X cubed over x minus four to the one
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half minus one times the derivative Of XQ over X
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-4. Following the chain rule. So this is
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1/2 times XQ over X-4 to the negative 1/2
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times deserve a tyvek ocean of two polynomial. So
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is three X squared times six minus four minus x
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cubed times one over x minus four square. So
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this is 1/2 times. This can be written as
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one over the square root of X cubed over X
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minus one. Then we can invert the fraction.
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So we get square root of x minus four over
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the square root of X cubed Times. In here
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we have a three x cubed-12 minus 12 mm
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X x square minus X cubed over X-4 sq
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. Mhm. And so this is 1/2 times X
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-4 to the one half times these can be written
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as X correct effects because excuses exports and sex.
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And we separate the square root and we get this
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times two x cubed-12 x square over Eggs-4
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Squared. Yeah. And this is one half times
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X minus forced to the one half over X times
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X to the one half times uh 22 x square
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. Common factor of x minus six Over X-4
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square. And here we can simplify things here.
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Okay, so we can say this is the to
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cancel out This sex too. In fact we can
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let it as three houses better simplify with this square
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and this X-4 to the one. How with
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this one. So we get in moderator X two
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D two minus two thirds minus three. Third three
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half. Sorry That is one x. M5 with
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00:21:11.380 --> 00:21:15.009 A:middle L:90%
his ex and ex over X to the one half
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is excellent one house. So this is squared of
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X X-6. And in the denominator will have
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X-4 to the 3/2. That's our derivative.
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Finally. Yeah. Okay. and is zero when
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00:21:47.640 --> 00:22:03.240 A:middle L:90%
X zero or eggs equals six? It is not
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possible. Yeah That it's equal zero Because eggs got
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to be greater than four. Mhm possible at X
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equals zero because okay then X equals six. It's
234
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the only critical point. Mhm. So the candidate
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to have for the function of to have a minimum
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to know what kind of critical point this is.
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00:22:55.880 --> 00:22:59.140 A:middle L:90%
We get to find the second derivative if we do
238
00:22:59.140 --> 00:23:06.579 A:middle L:90%
that we get that second derivative of F. At
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00:23:06.579 --> 00:23:18.240 A:middle L:90%
any X is equal to 12 over mhm. Square
240
00:23:18.240 --> 00:23:23.450 A:middle L:90%
root of x times x-4 to the fifth.
241
00:23:30.039 --> 00:23:33.089 A:middle L:90%
And then the second derivative at the critical.6.
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00:23:33.099 --> 00:23:41.069 A:middle L:90%
Give us 12 over Square root of six Times 6
243
00:23:41.069 --> 00:23:45.670 A:middle L:90%
-4 to the 5th which is a positive quantity.
244
00:23:47.839 --> 00:24:02.460 A:middle L:90%
So for X equal, see ffx has minimum value
245
00:24:07.440 --> 00:24:08.799 A:middle L:90%
. And that's what we're looking forward. There is
246
00:24:08.809 --> 00:24:15.380 A:middle L:90%
eggs Equal six. And for that why is a
247
00:24:15.380 --> 00:24:23.089 A:middle L:90%
minimum in terms of X. And so uh if
248
00:24:23.099 --> 00:24:32.829 A:middle L:90%
x equals six we can find why you seen This
249
00:24:32.829 --> 00:24:34.490 A:middle L:90%
relationship here. So why is the square root of
250
00:24:34.490 --> 00:24:48.470 A:middle L:90%
execute over X-4? So the corresponding value minimum
251
00:24:48.470 --> 00:25:08.529 A:middle L:90%
value of F of y? Yes, the square
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00:25:08.529 --> 00:25:14.559 A:middle L:90%
root of X cube six to over six minus four
253
00:25:18.839 --> 00:25:26.660 A:middle L:90%
right here, That is six scores of six over
254
00:25:27.240 --> 00:25:32.150 A:middle L:90%
six final fours two squirts of two and 63 times
255
00:25:32.150 --> 00:25:33.819 A:middle L:90%
too, so we get six quarts of three times
256
00:25:33.819 --> 00:25:38.960 A:middle L:90%
square root of two over scores of two and so
257
00:25:47.339 --> 00:26:02.569 A:middle L:90%
six square to three, which is about been 10
258
00:26:02.569 --> 00:26:17.380 A:middle L:90%
point 39 23 inches. So they see the minimum
259
00:26:17.390 --> 00:26:23.380 A:middle L:90%
value of why and It is attained when x equals
260
00:26:23.380 --> A:middle L:90%
6".