WEBVTT
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let's sketch the graph of the function F of x
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equal to minus one third time sex For X greater
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than or equal to-2. And we will use
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that graft to find the absolute and local maximum and
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minimum values of the function. So we have 2
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-1 3rd time sex for eggs on the interval or
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X in the interval from negative to to the right
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. That is up to plus infinity. Close at
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-2. I'm gonna go this line that is determined
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completely by two points. So uh X equals negative
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two. The image is two minus one third times
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negative two, which is two plus two thirds.
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And that is 33 times two is six plus two
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is eight thirds. So the image at-2 is
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8/3. And the other the other point we can
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use is for X equals zero. We get to
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so it's zero. The images to is here.
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And this is the image of negative to which we
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draw with uh feel red circle and then the function
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decreases all the time to the right without any bound
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. So um with this behavior of the functions line
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in fact we can talk about a little bit before
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giving the solution to the problem that this graph can
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also be obtained by some transformation to the identity function
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. In fact we have the identity function. It
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is. Then this function is multiplied by 1/3.
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That makes the function tilt to the right a little
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bit because now the image of one is one third
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. So we have some kind of turn to uh
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the line is closer to the X axis. Now
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then we have negative that is a reflection with respect
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to the X axis, something like this. And
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then after that we shift really displace the graph up
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, achieved the graph up two units. So we
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get this or less and that's it. That's what
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we found here this line. So there's another way
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. But each time we have a craft which is
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functions a linear function. We can throw it the
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graphics line and we can write just determining two points
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. Okay so um so now if we look carefully
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at the graph we can see that we have an
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absolute maximum value Equal to 8/3 attained at-2 which
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is this value here which is this value here you
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see the highest point on the graph and it's included
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in the graphic of excellent attitude is included in the
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domain we are considering so we can stay here.
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That gap has Yeah it has an absolute maximum value
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eight thirds and that value of course At x equals
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-2. There is no local maximum. Okay,
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that's because at any point of the under graph we
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have images that are greater and less than the image
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at the point. So there's no local maximum.
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Yeah and in the case of the answer of maximum
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it is not local because we had no crafted left
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in fact. Okay so we respect to the minimum
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we had no outside the minimum because this graph is
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decreasing all the time to the right of the values
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effects that is, there is no lower bound to
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the graph. So I have has no absolute but
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also local. There is no local minimum because for
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any point in the draft there are points with greater
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images that are greater or smaller than the image of
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the point, so it has no absolute or local
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minimum. Told the only thing that this graph has
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is an absolute maximum value of eight thirds. Which
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of course at Mexico native to If we had considered
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this graph open on negative to that is if this
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point here is not included in the graph, the
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function we would have no external at all, but
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in this case is included, so it's the absolute
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maximum value, that's it for this function.