Answer to Question #50714 in Differential Equations for tinn

Question

How to find maxima and minima. Process
1. to find x where dy/dx=o. let the x =a
2. then find d2y/dx2
3.if d2y/dx2 at x=a >0 then we will find minima for x=a
4.if d2y/dx2 at x=a <0 then we will find maxima for x=a
5.if d2y/dx2 at x=a =0 then we will have to proceed to the d3y/dx3
6.if d3y/dx3 at x=a in not eqal 0 , we will not find any maxima or minima.
7.if d3y/dx3 at x=a equals 0 then we will proceed to d4y/dx4. and if d4y/dx4 at x=a >0 then it's minima and if < 0 then maxima.

Please explain the rule 6 above. Why will we not find any maxima or minima in that case? Please explain with figure and example.