2nd Derivative Test conquered?

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SUMMARY

The discussion introduces a novel approach to applying the second derivative test for identifying extrema in functions. The proposed method involves finding a point 'a' where the first derivative f'(a) equals zero, followed by determining the first non-zero nth derivative f^n(a). If n is odd, the point is an inflection point; if n is even, further analysis is required. Additionally, the sign of f^n(a) indicates whether the point is a minimum (f^n(a) > 0) or a maximum (f^n(a) < 0). This method is validated through visualization using Taylor series expansion.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives
  • Familiarity with Taylor series expansion
  • Knowledge of critical points and their significance in function analysis
  • Basic skills in mathematical notation and terminology
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  • Research the Taylor series expansion and its applications in calculus
  • Study the properties of odd and even derivatives in function analysis
  • Explore advanced topics in calculus, such as higher-order derivatives
  • Learn about inflection points and their role in graphing functions
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Students and educators in mathematics, particularly those focused on calculus and function analysis, as well as anyone interested in innovative approaches to derivative tests.

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?:cool:
 
Physics news on Phys.org
Yes, obvious, if you visualise the function from its Taylor series expansion:

f(x) = (x-a)^n f^n(a)/n! + ...
 

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