WEBVTT
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let's find the absolute minimum and maximum values of the
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function F of X equals X plus one over X
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. On the closed interval, 0.24. Mhm.
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First thing we have noticed here is that this function
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is continuous on the interval 0.24. Because the only
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value of X for which we will have discontinuity of
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this function is X equals zero which is not included
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in this interval. So if it continues the interval
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is a close interval. So we know F attains
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its extreme values on that interval. And moreover,
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we know that those extreme values are attained either at
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the end points of the interval or at critical numbers
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of the function. So we got to find the
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critical numbers of F. And we start by calculating
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the first derivative of F which is equal to one
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-1 over x squared Mhm. And we can see
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that this relative is defined At every point or value
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x on the close interval 0.24. And for that
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reason the only critical numbers of these function F are
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those values of X for which the first serve,
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A T V is equal to zero. Then we
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gotta start by solving this equation after relative equals zero
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. And that's equivalent to the equation one minus one
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over X squared equals zero. Because the first derivative
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of F is given by this expression, this equivalent
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also to one over X squared circle one which also
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equivalent to X square eagle one. And from this
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we know that X can be equal 2, 1
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or-1. Yeah, but X equal to negative
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one is not in the close interval from 0.2 24
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. Then the only critical number of F In the
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intervals Europe 24 is x equal one. So we
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got to evaluate the function at this spiritual point x
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equal one. And at the end points of the
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interval 0.2 and four among those three values are the
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extreme values of functions. So Let's start by evaluating
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f at 0.2. That is 0.2 Plus one over
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0.2 which is equal to 0.2 plus Uh 3.2 is
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to over 10. So this will be 10/2 is
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the reciprocal of that number. And that's five.
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So we got 5.2. Now F at four which
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is c right in point of the interval is equal
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to four plus 1/4. Just four plus 0.25.
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And so we get for 0.25 and finally F at
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the critical number one is one plus 1/1 which is
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to And so we can see here that the maximum
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value of the function Over the interval 3.24 will be
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f at 0.2 which is 5.2. And the minimum
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value of the function over that interval is two Attain
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at X Equal one critical point. So we can
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now write the answer to the problem. Uh huh
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. The absolute maximum value of F In the interval
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0.2 four is 52. And that value of course
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at the left point of the interval AX equals 0.2
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. Yeah. And on the other hand we have
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that the absolute minimum value of f in the interval
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0.2 four? Yes two. And that value of
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course or happens at at the critical number X equal
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one. So that's the answer to the problem.
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And then we may make summary. Now we first
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noticed that this function attained its extreme value over the
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given interval because the functions continue over the interval and
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the intervals a close interval. Okay. We know
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moreover that uh the extreme values there is the absolute
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minimum and absolute maximum values of the function over the
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closed interval are occurring either at the end points of
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the interval or at critical points for critical numbers of
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the functions. So we got to find the critical
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numbers of two function for that. We calculate the
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derivative of F and resolve or solve the of equation
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F. First derivative of F equals zero. And
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we got in this case two solutions one a negative
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one. But we gotta stick with solution that is
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in the interval 0.24 in this case is X is
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equal one. And now we have a lot to
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function at this critical point X equal one. And
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at the end points of the given interval. And
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from these three values we found that The maximum,
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the absolute maximum value is 0.5 attained at The left
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and point of the interval 0.2. And the minimum
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absolute minimum value of the function over the integral ship
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into forest to add or retain it. X Equal
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one. Which is the only critical number of F
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in dangerous. And remember we know that this value
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X equal one is the only critical number of F
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in the interval. Sure 24 because the other possibility
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which is negative one, It's not inside or within
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dangerous Europe 24. So it's the only critical point
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. A number of F. N is in fact
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the point or value where the function attained its minimum
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or absolute minimum over the interval.