WEBVTT
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a formula for the derivative of a function F.
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Is given with that. We want to answer the
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question, how many critical numbers does F have this
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case function derivative of F is equal 2. 100
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coastline square of X. That over 10 plus X
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square in that fraction minus one. So the first
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thing we got to say is that preservative is defined
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everywhere is well defined because the denominator temp plus X
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square is positive for all real numbers, eggs.
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So after relative exists for all values of X.
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Inter real numbers. And if we're supposed to,
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the main of the function is the whole real numbers
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. That is there is no evidence to say the
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opposite. So we can say that the only critical
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numbers of F are those who deserve a tip is
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about zero. Okay. So Uh huh. We
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have to solve this equation indicative of F equals serious
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. Here we are. I'm going to say that
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we are assuming that the men of the functions the
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real numbers. Okay, So yeah, The equation
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of derivative equals zero. It's the same as 100
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co sign square of X Over 10 plus x square
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-1 equals zero. And does the same as 100
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. co sign square of x equal to 10 plus
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X square. That is because we put nothing one
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to the right of the equation. We get The
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fraction equal one and then the denominator pass to the
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numerator to the right of the equation and we get
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this and equivalents it's true because the expression 10 plus
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X square is never equal to zero. So we
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can divide by that number. And now this is
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our equation and we can say something here. That
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is if we want to find the critical numbers of
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the function F. We've got to solve these equations
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here and the number of solutions status equation will be
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the number of critical numbers of F. Then the
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number quantity of critical numbers. Uh huh. Of
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F is equal two. The number of solutions of
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the equation 100 co sign square, vax equal 10
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plus X square. But for this equation we can
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say something Is to follow it 10 plus X square
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in absolute value. First of all is equal to
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itself. That's because the expression 10 plus X squared
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positive for all real numbers eggs. So it's absolute
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value is the same expression. And now This is
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the absolute value of 100 co sign square of eggs
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. Maybe it's better if you write that the other
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way around is clearer. Maybe it is I'm saying
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this his coat is here and then this go here
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because we are taking in fact after the value both
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sides of the equation here that is this equation.
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This new equation with absolute valuable size. But here
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from here to here we know that the actual value
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the same expression because expression is positive. And now
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here this absolute value over here This is 100 absolute
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value of coastline of eggs square. And we know
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the absolute value of consign of eggs is less than
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or equal to want. So this is less than
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or equal to 100 for all eggs in the real
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numbers. Then any solution of these equation in the
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solution of mhm 100 co sign square of X equal
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templates. X square must satisfy the relationship. 10
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plus X 100 less than or equal to 100.
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Yeah but we also have the X square is less
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than are able to tempt plus X square because is
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the same expression here plus a positive number. So
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this is true. And if we put this inequality
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over here we get that X Square is less than
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or equal to 100. That is the absolute value
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of X Taking Square roots. It's necessary to 10
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. So any solution of mm 100 co sign square
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of X equals 10. The six square. That
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is any critical number of function. Mhm must uh
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satisfied Absolute value of X. place several equal to
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10. That is all the routes we can find
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our there and there is another information about the function
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uh about discretion. The equation we have here is
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that any value X which is a solution to this
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equation. Give us another solution which is negative X
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. Because co sign of X and X square are
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even functions. So if in all the positive solutions
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we will have all the solutions by taking the opposite
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sign of the positive solutions. But in fact at
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this moment we can either use if we want a
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numerical method to solve the equation. But in this
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case we only want the number. So we can
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do a graphical inspection of the function. 100 Coulson
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square of x minus x squared minus stan. And
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that is we want to see intergraph the solutions of
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these equation. You go to Syria, there is
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the roots of this function here and we have don't
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describe in Medlab and the result I'm going to show
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you right now This is the graph of the function
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coast, 100,000 square of X minus X squared minus
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10. And you can see effectively that the function
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is even and even function if we have all these
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roots of the function, this is the X axis
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here. So we have one group here, another
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one here. Another here here. Yeah. And
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two more here. So that we have 1234567.
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Now we have the same quantity to the left at
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zero because it's an even function to here. 34
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five C. Except So we have 14 with that's
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the final answer to the problem. So to function
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100 coastline square of eggs minus X squared. Afghanistan
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has 14 roads. And so the equation 100 co
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sign square of X equals X squared plus tin has
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14 solutions. That is F has 14 critical numbers
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. And that's because that's because we showed that uh
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the equations after every difficult, serious you could into
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this equation and that is equivalent to the root of
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this function. So that's why the other thing we
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noticed here was that any solution has to satisfy this
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equation, meaning that all solutions got to be a
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Get to be inside the interval for negative 10-10,
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that is outside of that interval. There are no
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woods of the equation. That is why we only
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need to see the graph between negative 10 and 10
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because we assure outside that there is no other solution
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to the equation. And because the derivative exists for
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all X in the real numbers, the only critical
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numbers are these solutions to this equation after relative people
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see. And then we have 14 but 14 critical
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numbers.