Problem 46 Easy Difficulty

A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?

$ f'(x) = \frac{100 \cos^2 x}{10 + x^2} - 1 $

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Wen Z.

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Amrita B.

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Chris T.

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Video Transcript

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