The fourth derivative test

In summary, The fourth derivative test states that if the fourth derivative of a function is equal to 0 at a critical point, then the point can be classified as a minimum, maximum, or horizontal inflection point based on the sign of the eighth derivative. This test is applied to functions with a second derivative equal to 0. It is also possible to apply this test to higher order derivatives, such as the eighth derivative for functions like f(x)=x^5. However, there is no specific "eighth derivative test" and the general idea is to keep taking derivatives until a non-zero one is found, with odd derivatives indicating horizontal inflection points and even derivatives indicating maximum or minimum points.
  • #1
KLscilevothma
322
0
I was given a remark and an example in my notes.

Remark: If f ''(xo)=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?
 
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  • #2
I don't recall there being a 4th derivative test.
 
  • #3
It looks as if you're applying the second derivative test to a second derivative.
 
  • #4
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 (+).
 

What is the fourth derivative test?

The fourth derivative test is a mathematical method used to determine whether a critical point of a function is a local maximum, local minimum, or saddle point. It involves taking the fourth derivative of the function and evaluating it at the critical point.

When is the fourth derivative test used?

The fourth derivative test is typically used when other methods, such as the first or second derivative tests, are inconclusive in determining the nature of a critical point. It can also be used to verify the results obtained from other tests.

How do you perform the fourth derivative test?

To perform the fourth derivative test, you first need to find the critical points of the function by setting its first derivative equal to zero. Then, take the fourth derivative of the function and evaluate it at each critical point. If the fourth derivative is positive, the critical point is a local minimum. If it is negative, the critical point is a local maximum. If it is zero, the test is inconclusive.

What are the limitations of the fourth derivative test?

The fourth derivative test can only be applied to functions that are four times differentiable. It also does not provide information about the behavior of the function outside of the critical point, so it should be used in conjunction with other tests for a more complete analysis.

Are there any real-world applications of the fourth derivative test?

Yes, the fourth derivative test is commonly used in optimization problems in fields such as engineering, economics, and physics. It can also be used in analyzing the stability of systems in control theory.

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