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Math Help - Nature of stationary point

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    Nature of stationary point

    Find the coordinates of the stationary points on the curve y=x(x-2)^4. Determine the nature of the stationary points.

    I am able to find the stationary points are at x=2 and x=0.4

    When x=0.4, the second derivative of y=-20.48, hence it is a max point.

    When x=2, the second derivative of y =0, which means it is point of inflexion. However, when I plotted the graph of y, I realise that it is a minimum point. Which is correct?
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    Quote Originally Posted by yeoky View Post
    Find the coordinates of the stationary points on the curve y=x(x-2)^4. Determine the nature of the stationary points.

    I am able to find the stationary points are at x=2 and x=0.4

    When x=0.4, the second derivative of y=-20.48, hence it is a max point.

    When x=2, the second derivative of y =0, which means it is point of inflexion. However, when I plotted the graph of y, I realise that it is a minimum point. Which is correct?
    The second derivative test is INCONCLUSIVE if the second derivative = 0 at that stationary point. You need to check the gradients at points close to the stationary point and show that it goes from positive to negative or vice versa.


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    Quote Originally Posted by Prove It View Post
    The second derivative test is INCONCLUSIVE if the second derivative = 0 at that stationary point. You need to check the gradients at points close to the stationary point and show that it goes from positive to negative or vice versa.


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  4. #4
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    The fact that the second derivative is 0 does NOT mean it is an inflection point. An inflection point is where the second derivative changes sign. Of course, for a smooth function, that can only happen where the second derivative is 0 but the converse is not necessarily true.
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