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Math Help - Find points of inflection

  1. #1
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    Find points of inflection

    The curve y= 2x/(1+x^2) has

    1. exactly three points of inflection seperated by a point of maximum and a minimum.

    2. exactly two points of inflection with a point of maximum lying between them.

    3. exactly two points of inflection with a point of minimum lying between them.

    4. exactly three points of inflection seperated by two points of maximum.



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    My working: I derived the above expression and found the derivative to be 0 at x=1,-1 . So i really dont know how can there be three points of inflection at all.
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  2. #2
    Senior Member Twig's Avatar
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    hi

    Hi

    Im in kinda of a hurry, but I also just got one inflection point, x=0.
    I might have missed something..
    Attached Thumbnails Attached Thumbnails Find points of inflection-graph.jpg  
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  3. #3
    Moo
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    Hello,

    The second derivative turns out to be :
    \frac{4x(x^2-3)}{(1+x^2)^3}=\underbrace{\frac{4}{(1+x^2)^3}}_{p  ositive} \cdot x(x-\sqrt{3})(x+\sqrt{3})

    At an inflection point, the second derivative changes its sign.
    If you make a simple table of signs for x(x-\sqrt{3})(x+\sqrt{3}), you will see that there are indeed 3 inflection points.

    The points where the first derivative is 0 are called "critical points". They're not the same as inflection points !
    However, finding the first derivative and the critical points will help you answer the parts dealing with maximum/minimum.
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  4. #4
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    Hi Moo.

    Is it possible for you to point out the three inflection points in the above graph? Inflection points as I understand are points where the curve changes from concave to convex or vice-versa . Am i correct in this?
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    Moo
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    Quote Originally Posted by champrock View Post
    Hi Moo.

    Is it possible for you to point out the three inflection points in the above graph? Inflection points as I understand are points where the curve changes from concave to convex or vice-versa . Am i correct in this?
    Yes, you are correct in this. And what I said is an equivalent definition.

    Here is the graph.
    The inflection points are approximatively where the red circles are.
    The -sqrt(3) and sqrt(3) inflection points are not easy to see because the change concave-convex is not very clear (though I found it easy to find it for -sqrt(3))

    Did you try that signs table for x(x-sqrt(3))(x+sqrt(3)) ? Actually, a point where a function changes sign is a point where the function is 0. That's logic ^^
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