High School Find Local Max/Min: 2nd Derivative=0

[greg_rack]

How do I distinguish between a point of local maxima or minima, when the second derivative in that point is equal to zero?

 

[snorkack]

"Stationary point" means that first derivative is zero and you specify that second also is.
In that case, if the first nonzero derivative is odd (third, fifth etc.), it is neither maximum nor minimum but a stationary inflection point. If the first nonzero derivative is even (fourth, sixth etc.), the point is minimum if that derivative is positive and maximum if negative.

 

[greg_rack]

"Stationary point" means that first derivative is zero and you specify that second also is.
In that case, if the first nonzero derivative is odd (third, fifth etc.), it is neither maximum nor minimum but a stationary inflection point. If the first nonzero derivative is even (fourth, sixth etc.), the point is minimum if that derivative is positive and maximum if negative.

Got it!
That seems quite counter-intuitive... is there a way to demonstrate it "empirically", or is there just an analytical way to do it?

 

[snorkack]

The functions where first nonzero derivative is an odd one behave like odd powers of x and don´ t have extrema. It is a stationary inflection point. The functions where first nonzero derivative is even one behave like even powers of x, and do have extrema. When second derivative is specified zero, but first nonzero derivative is even one then the extremum is a flattened extremum, but an extremum nevertheless.

 

[suremarc]

This is more or less what snorkack said, but if you have some function $f(x)$ which has zero $k$th derivatives for $k\lt n$, you have $f(x)\approx \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$. The higher order terms are dominated by the $n$th term if $x$ is close enough to $x_0$. So basically, in some small enough neighborhood of $x_0$, the behavior of $f(x)$ reduces to the behavior of the power function $\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$.