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Math Help - Find the points on inflection and discuss the concavity of the graph of the function?

  1. #1
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    Find the points on inflection and discuss the concavity of the graph of the function?

    I need to find the points on inflection and discuss the concavity of the graph of the function:

    f(x)= (x+1)/(sqrt(x))

    I know that points of inflection require setting the second derivative equal to zero and finding where it is undefined.

    Here is my work so far...
    f(x)=(x^(-1/2))(x+1)
    f(x)=x^(1/2)+x^(-1/2)
    f'(x)=1/2x^(-1/2) -1/2x^(-3/2)
    f''(x)= -1/4x^(-3/2) +3/4x^(-5/2)

    My problem is that when I find the zeros of f''(x) I get x=3. That makes sense. But when I see where it is undefined, it is undefined from (-infinity,0).

    How can I use this information to find points of inflection and where the graph is concave up/down?

    Thanks so much!!!
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  2. #2
    MHF Contributor
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    Forget the derivative for a moment. What is the Domain of f(x)?

    Are you really surprised that the second derivative doesn't exist for x <= 0?

    You can check the sign of the first derivative on both sides of the point of inflection to see the concavity.
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