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Math Help - The second derivative test

  1. #1
    No one in Particular VonNemo19's Avatar
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    The second derivative test

    Could someone please explain to me why the second derivative test works. My professor taught me how to use it, but not why it works. How can the second derivative tell whether or not a graph is cocave up or down?
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  2. #2
    Super Member Random Variable's Avatar
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    In an interval in which a graph is concave up, the slope of the tangent lines to the graph are increasing (i.e. f''(x)>0). And in an interval in which a graph is concave down, the slope of the tangents lines are decreasing.
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Random Variable View Post
    In an interval in which a graph is concave up, the slope of the tangent lines to the graph are increasing (i.e. f''(x)>0). And in an interval in which a graph is concave down, the slope of the tangents lines are decreasing.
    So simple. Thank you.
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  4. #4
    Member McScruffy's Avatar
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    As posted the second derivative itself can tell you if the function is concave up or down, but the second derivative test can tell you f(c) is a local maximum or minimum. As stated if f is concave up (f''(x)>0) near c, then f lies above its horizontal tangent at c. Therefore, f(c) would be a local minimum, and similarly if f was concave down near c.
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