How Can the Second Derivative at a Maximum Be Zero?

[Mathsforum100]

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.

 

[DonAntonio]

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.

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

DonAntonio