WEBVTT
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let's find the absolute minimum and absolute maximum values of
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the function F of X equals extra negative two times
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natural algorithm of X On the closed interval from 1/2
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24. The first we know that this function is
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continuous and then given interval because the point example series
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not there. And in fact the natural algorithms are
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defined for positive values of X. But here we
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have an interval which is completely on the positive values
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of X. So uh the functions continue there.
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And because the internet is closed, we know that
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the function obtains its extreme values on that interval and
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the points where the extreme values of F. R
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attain it can be either the end points of the
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interval or the critical numbers of F. So we
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start by calculating the first derivative of F. And
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that's a product here we have X squared X.
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To the negative two times Natural algorithm of X.
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We can see it also as steady as a fraction
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as coaching last like natural rhythm of X over X
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squared. But let's see it like a product.
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And then we apply the product rule for differentiation.
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So so we get the derivative of extra negative two
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is negative two X. To the negative three times
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. And natural rhythm of X plus X to the
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negative two times the derivative of natural rhythm of X
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which is one over X. So we get negative
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two. Natural rhythm of X over X. Q
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. It's the first term here plus one over X
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. Q. And that becomes one minus two.
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Natural rhythm of X over excuse. So that's the
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derivative and preservative is not defined at zero that does
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not exist at zero. And in fact it's also
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defined only for positive values of X. So zero
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is not a critical number of F. Because even
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though the derivative does not exist there and that Value
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zero is not in the domain of the function.
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So We can say that after evocative at zero does
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not exist but zero is not in the domain of
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F. So it is not critical number of F
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. So we've got to solve the equation. First
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derivative F equals zero. And that's the same as
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the equation one minus two. Natural rhythm effects over
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excuse. And we are supposing on this equivalence is
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here that X is positive. So that we can
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take the natural algorithm. So this is equivalent to
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the equation. One runs to natural algorithm of X
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equals zero. Which is equivalent to natural rhythm of
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X equal to one half. Because the exponential function
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is injected function. We get for the positive numbers
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. Yet this equivalent to E. To the natural
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algorithm of Mexico to eat to the one half.
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And this girl into because the natural algorithm and the
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exponential function are inverse to each other. We get
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x equal into the 1/2 which is quite a bit
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of E. And if we calculate that it's about
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um Squares of E. is about 1.64 1.64 87
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87 1 2- 1- seven. Mm.
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Oh seven. Okay. This is uh so this
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is the only uh x equal spirit of E is
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the only critical number. Let's see in fact that
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this number is in the interval one has four because
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one has 0.5. So between 0.5 and four is
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the number 1.6487. So they see a critical number
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of F In the interval 1/2 4. In fact
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, this is the only critical number of functions in
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its extended domain which is a positive numbers. Okay
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, so we have this and now we can find
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the extreme value of the function by evaluating the function
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at this critical number and at the end points of
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the interval That is 1/2 and four. So what
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we have here is f at 1/2 is equal to
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let's see, Excellent. Native to times. Natural
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algorithm of eggs. That's the same as natural rhythm
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of 1/2 over one have square. And remember the
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natural rhythm of fraction is the natural rhythm of the
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numerator minus. And natural rhythm of the denominator.
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There's a property of the logarithms And 1/2 square is
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1/4. Natural rhythm of 10. So we get
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negative. Natural algorithm of two over 1/4 in that
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negative. Yeah, that's negative. Uh for natural
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rhythm of two, which is about negative value is
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about negative. 2.772 five. Yeah. 887.
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Let's put a little bit too left. That's about
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2.7725887 2224 three times two. Yeah. Okay.
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Now f at the other end point that is at
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the writing point is for And that is natural rhythm
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of four over for square. Remember four here is
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naturally almost two square over uh 16. And these
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applying the properties of the law Rhythm, we get
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to natural rhythm of to over 16. And that's
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the same as Natural algorithm of 2/8. And that's
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about 0.08 664. Okay. And that's the end
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points. Now the critical number is scores of E
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. And the images. Natural rhythm of this word
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of E. Over mm. Spirits of E.
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Square. And here we get one half over mm
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That is one over to eight which is about To
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your.18 39397 206. So between these values is
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the absolute minimum and the actual um and similar values
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of the function. And we recognize Isley that The
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absolute maximum value of the function is 0.1839397206. Which
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is one over to E. And it is the
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image of the critical number, scores of E.
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And the minimum value is as a minimum value of
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the function is negative four. Natural rhythm of two
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which is about 92 7725887 to 2 to four Which
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is the image of the left hand.1 half.
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So we can write the answer to the problem.
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You can say that the absolute minimum value of F
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on the interval one half four. Yes We said
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-4 naturally the most to which is about negative 27725
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sorry two point 7725 887 2224. And that minimum
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value of course at um The left hand.1 half
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. And for the on the other hand the absolute
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maximum value of F On the close into one have
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four is 1/2, one over to E. Which
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is about zero 18 3939 7-06 72. It's your
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six which of course at um the critical number X
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equal squares of E. Which in fact is more
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or less equal to 1.64 8721 2707. So this
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is if I know the answer to the problem and
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we remember what we did here. We first recognized
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that this function is continuous Over the close intervals so
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it attains its extreme values at that interval. And
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knowing that those uh stream values are obtained either on
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the end points of the interval or critical numbers of
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the function, we calculate the first derivative of F
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and see if there are values in the domain where
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the derivative is not defined. It's not the case
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here. There is even though zero for example is
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evaluate the preservative does not see exist that value serious
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not in the domain of function. So it is
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not a critical number. And we saw that in
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fact the function and its derivative are defined for positive
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value of X serious not included. So but the
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given interval is completely in the interval or is contained
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in the interval of positive numbers. So the derivative
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exists for all values of vaccine that into this case
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. So the only critical numbers here are those for
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which differ serve a tip zero. We state that
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equation and then we solve it to find that the
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only critical number of the function in the interval one
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half four is X equals scores of E, which
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is about 1.6487 to 1 to 7 37. Then
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we evaluated the function at the end points of the
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interval and at the critical number squared E. And
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we found that the Largest value is 0. 1
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over to me which is about 0.28 0.1839397- six
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. And it occurs at the critical number of records
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of E. And the smallest of the values is
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negative four times natural rhythm of two, which is
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about negative two.7725887- 2- four. And
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it accords at the left end point of the interval
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one half. And so we have found the absolute
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minimum and excellent maximum value, the function over to
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give an interview.