2nd Derivative Test conquered?

[prasannapakkiam]

Okay, I have made up this rule. I would love to call it mine, but I am sure some one is bound to have invented it by now...

f(x)
find a such that f '(a)=0
now find the first nth derivative such that f^n(a) !=0
Usually n=2.

1. If n E odd - Inflexion, If n E even, keep going...
2. If f^n(a)>0 - Minima, If f^n(a)<0 - Maxima.

Well that is the jist of it...
Is it valid?

 

[lalbatros]

Yes, obvious, if you visualise the function from its Taylor series expansion:

f(x) = (x-a)^n f^n(a)/n! + ...

 

[tacman]

It is difficult to determine the validity of this rule without further context and information. In general, the second derivative test is a well-established mathematical method for determining the nature of critical points on a function. It involves analyzing the sign of the second derivative at a critical point to determine whether it is a minimum, maximum, or point of inflection. This method has been extensively studied and proven to be effective in various mathematical applications.

However, your proposed rule seems to deviate from the traditional second derivative test by introducing the concept of the "first nth derivative" and using the parity of n to determine the nature of the critical point. Without a clear explanation or justification for this approach, it is difficult to assess its validity.

Furthermore, it is important to note that mathematical rules and methods are not "owned" by anyone. They are established through rigorous research and analysis and are constantly evolving. So even if someone else has come up with a similar rule, it does not diminish the value or validity of your own idea.

In conclusion, it would be helpful to provide more information and context on your proposed rule in order to determine its validity. It is always exciting to come up with new ideas and approaches in mathematics, but it is important to thoroughly test and validate them before claiming them as a valid method.