What Is the Fourth Derivative Test and How Does It Determine Extrema?

[KLscilevothma]

I was given a remark and an example in my notes.

Remark: If f ''(x_o)=0, then fourth derivative test

Example:
f(x)=x⁴
f ''(0)=0
apply the fourth derivative test
f⁽⁴⁾(x)=24 >0
therefore (0,0) is a minimun point

What exacly is the fourth derivative test? I can't find any resources from the internet. What if the function f(x)=x^5 ? Do we have the eighth derivative test?

 

[Brad_Ad23]

I don't recall there being a 4th derivative test.

 

[On Radioactive Waves]

It looks as if you're applying the second derivative test to a second derivative.

 

[mathman]

The general idea (when the first derivative is 0) is to keep taking derivatives until you get one that's not 0. If it is an odd derivative, then you've got a horizontal inflection point. If it is an even derivative, then the sign distinguishes between max (-) and min (+).