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Math Help - analysis of rational function

  1. #1
    Senior Member euclid2's Avatar
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    analysis of rational function

    for: f(x)=\frac{(x-1)^2}{x^2+1} determine:

    1/x and y intercepts
    2/co-ordinates of all critical values
    3/classify the critical values using the second derivative test
    4/determine the coordinates of all possible points of inflection
    5/check to make sure there is a change in concavity at these points
    6/determine all increasing and decreasing intervals
    7/determine the intervals of concavity
    8/determine the equation of the horizontal asymptote
    9/provide a sketch and label all parts of the graphical analysis determined above

    so for f'(x) i get f'(x)=\frac{2(x^2-1)}{(x^2+1)^2} and for f''(x) i get f''(x)=\frac{-4x(x^2-3)}{(x^2+1)^3} both of which i got using the quotient rule, although very unfamiliar as to how to work out each step
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  2. #2
    Master Of Puppets
    pickslides's Avatar
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    Well here's some advice

    Quote Originally Posted by euclid2 View Post

    1/x and y intercepts
    x-intercepts make y=0, what do you get?

    y-intercepts make x=0, what do you get?

    Quote Originally Posted by euclid2 View Post

    2/co-ordinates of all critical values
    Find where f'(x)=0

    Quote Originally Posted by euclid2 View Post

    3/classify the critical values using the second derivative test
    for the points found in the previous question, plug them into f''(x) . Then if f''(x)>0 \implies \text{min}, f''(x)<0 \implies \text{max}, f''(x)=0 \implies \text{pt of inflection}

    Quote Originally Posted by euclid2 View Post

    4/determine the coordinates of all possible points of inflection
    I think I have already answered that in my previous explanation

    Quote Originally Posted by euclid2 View Post

    5/check to make sure there is a change in concavity at these points
    Pick a point either side of each critical point. Then check the gradients of these points with f'(x), they should have opposite signs. This implies a change in concavity.

    That is enough for now, have a go, let me know what you think.
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