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Second derivative at a maximum

  1. Jun 28, 2012 #1
    The second derivative at a maximum is either negative or zero. Can you explain how it can be zero? There can't be a 'plateau' at the maximum or it would not be a point. I cannot imagine graphically how the second derivative at a maximum can be zero. Before the maximum, the gradient is positive. After the maximum it is negative. So the gradient is decreasing.
  2. jcsd
  3. Jun 28, 2012 #2

    The function [itex]\,f(x)=-x^4\,[/itex] fulfills
    $$f'(0)=f''(0)=f'''(0)=0\,,\,f^{(iv)}(0)<0$$and, of course, it has a local maximum at [itex]\,x=0[/itex] ...

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