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Show that the curve $y=(1+x) /\left(1+x^{2}\right)$ has three points of inflection and they all lie on one straight line.

the slopes are the same, all inflection points lie on the same line

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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Lectures

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Show that the inflection p…

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Show that the curves $y=e^…

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Show that if $ f(x) = x^4 …

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Show that if $f(x)=x^{4},$…

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Show that any cubic $f(x)=…

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Use the first and second d…

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Show that the curve $ y = …

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Show that $f(x)=x^{3}-3 x^…

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Consider the curve $x=y^{3…

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Find the inflection points…

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Show that $D=0$ for $f(x, …

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Linearizations at inflecti…

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Show that the general quar…

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A curve of the form $y=e^{…

Um Since we want to discuss about the inflection points, we need to figure out the second of narrative first. So this is this is this iconic narrative off. Why? I'm so he said the white double primary cause is zero. So you have three solutions. X equals to one X equals Dio, um, minus two class route off three. And the X equals two minus two miners. Fruit off three. Okay, so what? We can verify it off them. Our inflection point. So for example wait. We have the resolution. So we have forcibly intervals. So from minus infinity to minus two minus Wrote off three. Oh, so from minus two minus root off three to minus to pass route off three and from minus to pass. Wrote off 321 from one to infinity. So, over the first interval on the second narrative White up a prime ah will be negative Sitcom cave. Don't on the second to go by the book Primates Positives. It can keep up on the 30 in humble. Why double prime? We're being negative again. So it's conclave on the last interval. White double primates. Positives. It's can't give up. That means we have off these I inflection points. So your collection points at X equals toe one minus two Pross Brutal three and, uh minus two. Minus fruit off three. And they want oh, verified. Ah, off this inflection point lies on the same line. So when X equals 21 we have why it cost one when x supposed to minus root up to class three a minus to pass route off three. And why we will be ah, route off the re minus 1/8 minus four times. Route three in the last wise minus two minus. Food off three And the white value will be route minus root of three minus one over eight plus 4/10 of three verified off this point a be on the sea Nice on the same line So we just need to verify they have so each to off them had has the same slop. So, for example, the slope for a straight i a b ah, we will be root off three minus 1/8 minus four times wrote off three minus one and the miners divided by minus to pass route of three minus one and the one being verified. This is wonderful. And we took a something for point A and point to see. The result is also won over four The Muse, ABC, Our own the sin my

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