STATIONARY POINTS  
Analysis  
A function y=f(x) has a STATIONARY POINT at x_{o} when f'(x_{o})=0
In this case we know that the tangent to the curve is horizontal at x_{o} .
You have already seen that there can be a local maximum or minimum depending on the sign of f''(x_{o}), but when f''(x_{o}) is also equal to 0, there could be a point of inflexion.
So, what happens when successive derivatives are equal to 0 at x_{o}?
Look at the graph of the function y=x^{4}1 in the window and note that it has a minimum at x=0.




Look at the graph of the function y=x^{5} in the window. As you can see in the graph, f has a point of inflexion at x=0.

María José García Cebrian  
Spanish Ministry of Education. Year 2001  
Except where otherwise noted, this work is licensed under a Creative Common License