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Math Help - How to find the find local max, min and inflection points from an integral?

  1. #1
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    How to find the find local max, min and inflection points from an integral?

    I am trying to understand how to find the local max, min, inflection and critical points just by looking at the graphs from an Integral. Someone please help me.

    The question says f(x)= x^2-5x-6 and F(x)= integral (0 to x) f(t) dt.

    Find the critical points, local max, min and inflection points.

    Solution:

    Using the second FTC, I got

    F(x) = integral (0 to x) (t^2-5t-6) dt

    so F'(x) = x^2-5x-6
    and the graph of this is included at the bottom. So from the graph I can understand that the critical points are -1 and 6 since F'(x) is the derivative of the integral.

    But I don't know how to find local max,min and inflection points from this graph (given that it is the graph of the derivative of an antiderivative). Can anyone please help me find them/ explain them how to find it please?

    (I know that to find local max, min and inflection points, I use the second derivative of a regular function. But I am not sure how to find them using the 2nd FTC)

    SOMEONE PLEASE HELP!!!!
    Attached Thumbnails Attached Thumbnails How to find the find local max, min and inflection points from an integral?-graph.png  
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  2. #2
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    Quote Originally Posted by skboss View Post
    I am trying to understand how to find the local max, min, inflection and critical points just by looking at the graphs from an Integral. Someone please help me.

    The question says f(x)= x^2-5x-6 and F(x)= integral (0 to x) f(t) dt.

    Find the critical points, local max, min and inflection points.

    Solution:
    Using the second FTC, I got

    F(x) = integral (0 to x) (t^2-5t-6) dt

    so F'(x) = x^2-5x-6
    and the graph of this is included at the bottom. So from the graph I can understand that the critical points are -1 and 6 since F'(x) is the derivative of the integral.

    But I don't know how to find local max,min and inflection points from this graph (given that it is the graph of the derivative of an antiderivative). Can anyone please help me find them/ explain them how to find it please?

    (I know that to find local max, min and inflection points, I use the second derivative of a regular function. But I am not sure how to find them using the 2nd FTC)

    SOMEONE PLEASE HELP!!!!
    The graph shown is the graph of the derivative. A local max occurs a where the derivatve changes from positive to negative. A local min occurs where the derivative changes from negative to positive. An inflection point occurs where the function changes concavity. The function is concave up when the second derivative is positive (where the graph of derivative is increasing). The function is concave down where the second derivative is negative (where the graph of the derivative is decreasing). Hope that helped you.
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