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Find the critical numbers of the function.$h(p) = \frac{p - 1}{p^2 + 4}$

$p=1 \pm \sqrt{5}$ are the critical numbers

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Anna Marie V.

Campbell University

Kristen K.

University of Michigan - Ann Arbor

Samuel H.

University of Nottingham

Michael J.

Idaho State University

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Video Transcript

we want to find the critical numbers of the function asia be equal B -1 over p square plus four is a rational function. And we remember that the values into the main of a function for which the derivative of the function at that point does not exist Or exist in support zero. It's what we call the critical numbers. So we want to find points in the domain our numbers in the domain of the function for which the narrative that does not exist or these sequences zero. The first thing we got to notice here. So the main of the function age given here is the real numbers, sorry, serial numbers. And they because you think that we have a fraction the denominator piece square plus four is never got zero, which will be a point. The value of people with the denominator will be zero Or the values were able to be zero. Won't be on in the domain of the functions. So in this case those fines don't exist because P square Plus four is always greater than or equal to four, which is strictly positive. So it's never zero. Or saying another way we have a real number P square it that's positive or zero. And we add for will be a positive number. That cannot be zero. For that reason the domain of this function is the real numbers. And on the other hand, we have the derivative of this function is found by the derivative of Kocian. So we get derivative of the numerator respect. B is one times the denominator without derivative minus the numerator times derivative of the denominator respect to P is to be in this case over P square for four plus four. There's a denominator square. We simplify the numerator here, we get the square plus four minus two piece square. We are distributing this to be inside parenthesis. So we get negative to be square plus to be over Peace Square Last four. And then some square. And I was simplified a little bit. And then we get negative too. Sorry. Yeah. You have to know nothing square. Just one B squared minus two. P square is negative. The square plus to be that's four over Peace Square Plus four sq. Yeah. And these are derivative is defined at every real number because again we have a fraction but the denominator never vanished. So exists for any E in the real numbers. So we are not going to find political numbers that satisfies this part of the definition that is for which the derivative derivative does not exist because derivative exists for every for every serial number B. Then the critical numbers Okay. Of age are they're real numbers? Because of the main function is the real numbers. Forage Derivative of age is zero. Yeah. You have to be more precisely real numbers B. For which age derivative at P is zero. So we can only fine. Yeah. Critical numbers that satisfies this second part definition. Okay, So then Age derivative is zero. It's equivalent to say that this equation is zero here because that's the general expression of the derivative. So now the square Plus two p plus four over P sq was 4 sq0. And this is the same as I had the peace square plus two, P plus four equals zero. And this uh it was it's good to another analyze little bit If we have that this inspiration is equal to zero because we have a fraction here. And the numerator is just this expression here being this oppression because syria we will get here. Diffraction has zero over this number. That gives us zero because we know that number never be zero denominator for any value of P. And if we have that this fraction is able to zero. We passed the denominator to the right multiplying and that will give us the numerator equal to zero. So they are equivalent. In fact, sometimes it's a good idea to stop a little bit and think if the equivalence is we are writing our true or are well defined, well established. So in this case this is the situation. So now this is equivalent to saying that the square -2 p -40 because we only change signed both sides. And now we have a signal order Equation of polynomial of Degree two. So we can apply the formula now. Okay, so finding the critical numbers which we know in this case are those who satisfy this equation Is the same as solving this equation here. So we solve that equation and then we know that B is given by Uh two more or less squared off negative two square minus 41 Which is the coefficient of x squared times the independent term minus four Over two times a coefficient of peace square, that is two. How would that he was to more or less square to four plus because we have minus minus and 16. and this is the same as tube more or less coverage of 20 over to In 20. Remember is five times four. And we can separate the square Into the factors and we get two more or less where the five times 4-4 where the forest to. So we get two more or less to score to 5/2. We can separate the sum or subtraction in the numerator. Uh huh With the denominator. And we get one that is to hear over to more or less and two square to five over to see Square five mm. Then the critical numbers of function it should be equal P -1 over the square plus four. Uh huh. Are So we have one taking the plus side, one plus words of five And then thank you taking the negative sign with 1- groups of five. Mhm. We have found. And they are the only critical points because these are the only solutions to the second degree equation and the solutions of these on the second degree equations are the solution of the equation derivative of agent, considering because these are equivalent expressions and we saw that the derivative exists everywhere. So this could be the only critical points. Because the other option that there is this point in the domain where derivatives that does not exist is not possible in this case. So the answer, the final answer is that this function has two critical points only, and they are one plus 4 to 5 and one minus 35.