Exception to second derivative test? (Or am I doing something wrong?)

In summary, the conversation discusses the 2nd derivative test for finding the nature of stationary points and how it does not seem to work for a specific equation. The equation y=(x^2-1)4 is given and the stationary points are found using normal calculus. However, when substituting in the values, it is discovered that the points are not stationary points of inflection as initially thought. The necessary but not sufficient condition for inflection points is also mentioned. The conversation ends with a thank you for the clarification.
  • #1
Dramacon
14
0

Homework Statement


I'd always used the 2nd derivative test for the nature of stationary points. But I came across this equation in one of my exercises, for which the test doesn't seem to work at all.

Find the stationary points of: y=(x^2-1)4, stating the nature of each.

Homework Equations


Using normal calc: the stationary points are at (-1,0), (0,1) and (1,0)

The Attempt at a Solution


Although the double derivative works for when x=0, (local max)

When I sub in the values x=1 or x=-1, the value I end up with is 0: suggesting that these points are stationary points of inflection, when they are not (they're actually local minima).

Please help. :)
 
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  • #2
Hi Dramacon! :smile:

From http://en.wikipedia.org/wiki/Inflection_point#A_necessary_but_not_sufficient_condition :wink:

A necessary but not sufficient condition

If x is an inflection point for f then the second derivative, f″(x), is equal to zero if it exists, but this condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection. (An example of such a function is y = x4).​
 
  • #3
Ah, I see! Thank you! :)
 

1. What is the exception to the second derivative test?

The exception to the second derivative test occurs when the second derivative of a function is equal to zero at a critical point. This means that the test cannot determine whether the critical point is a minimum, maximum, or inflection point.

2. How does the exception to the second derivative test affect my analysis of a function?

If the second derivative test cannot determine the nature of a critical point, you will need to use other methods, such as the first derivative test or graphing, to analyze the function and determine the nature of the critical point.

3. Are there any other limitations to the second derivative test?

Yes, the second derivative test can only determine the nature of critical points on a continuous function. It cannot be applied to discontinuous or piecewise functions.

4. Can the exception to the second derivative test occur at more than one point on a function?

Yes, it is possible for the second derivative to equal zero at multiple points on a function. In this case, the test cannot determine the nature of all of these points.

5. Is the second derivative test always reliable in determining the nature of a critical point?

No, the second derivative test is not always reliable. It can only provide information about the nature of a critical point, but it cannot guarantee that the point is a minimum, maximum, or inflection point.

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