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Math Help - Second derivative = 0

  1. #1
    Member maxpancho's Avatar
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    Second derivative = 0

    When we're trying to figure out whether a critical point is a maximum or a minimum we can use second derivative test, and if we find that the point of our interest has second derivative equal to zero, this means that there can still be a maximum or a minimum at this point.

    Why is this the case?
    Last edited by maxpancho; August 20th 2014 at 02:07 PM.
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  2. #2
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    Re: Second derivative = 0

    Easiest to look at an example. Consider the two functions

    f(x) = x^4,  g(x) = - x^4

    Both have critical points at x = 0 yet both also have their second derivatives vanish at x = 0 so the second derivative test is inconclusive for this case. However, it's not hard to see that

    f \ge 0 and g \le 0

    so for f, it has a minimum at x = 0 whereas g has a maximum there.
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    Member maxpancho's Avatar
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    Re: Second derivative = 0

    Does someone have another example?
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    Re: Second derivative = 0

    Funny, so there are two distinct cases:

    1. the rate of change of a function is zero at a particular point but it is continuing to grow (functions like $x^2$)
    2. the rate of change of a function is zero at a particular point and the function sort of lingers at it, while x is at this value (and in the case of $x^4$ immediately starts increasing just after it gets past this point, where the slope is zer)
    Last edited by maxpancho; August 20th 2014 at 03:40 PM.
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    Re: Second derivative = 0

    Hello, maxpancho!

    When we're trying to figure out whether a critical point is a maximum or a minimum,
    we can use second derivative test. .And if we find that the point of our interest has
    a second derivative equal to zero, there can still be a maximum or a minimum at this point.

    Why is this the case?

    Let x = c be the critical point.

    If f''(c) > 0, the graph is concave up: . \smile

    If f''(c) < 0, the graph is concave down: . \frown

    If f''(c) = 0, the graph is neither concave up nor concave down.
    . . Using baby-talk, the graph "flattens" briefly at that point.
    The point could be a maximum, a minimum, or an inflection point.
    . . We use additional tests to determine which it is.

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  6. #6
    Member maxpancho's Avatar
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    Re: Second derivative = 0

    So this tells us that a parabola like $x^2$ doesn't flatten at zero, but something like... well, it seems that at least when we consider $x^n$, when $n \in \mathbb{Z} $, $n > 2$ the function flattens at zero. Interesting.
    Last edited by maxpancho; August 21st 2014 at 06:26 PM.
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