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

[Dramacon]

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. :)

 

[tiny-tim]

Hi Dramacon!

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

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).​

 

[Dramacon]

Ah, I see! Thank you! :)