The 1st derivative test
takes -local- information about $f'$, and
yields -local- information about $f$
Cool, but not too impressive.

The 2nd derivative test
takes -infintesimal- information about $f'$ and $f''$, and
yields -local- information about $f$
IE an infintesimal -> local theorem

These are easy to prove naively, just using derivative tests.

The general $n$th derivative test
(infintesimal -> local)
even
  if $f^{(k)}(a)=0$ for $0<k<2n$ and...
  ...$f^{(2n)}(a) > 0$
     then local min
  ...$f^{(2n)}(a) < 0$
     then local max
odd
  if $f^{(k)}(a)=0$ for $0<k<2n+1$
  and $f^{(2n+1)}(a) \neq 0$
  then neither local max nor min

Proofs:
- use fundamental theorem of calculus a bunch of times and then apply
  $1$st derivative test.
(great exercise in Fundamental Theorem of Calculus section!)

- use Taylor polynomials: function looks like $\pm (x-a)^n$,
  which obviously has max/mins when specified.

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generalizations:
the 1st derivative test is much less useful in 2 or more dimensions;
you can't say
``$f'$ is positive on one side and negative on the other'',
since punctured neighborhoods are -connected-!
The correct generalization is:
if the -gradient field- has a zero of positive order, then you're a
  local min, if of negative order, then you're a local max
  (or maybe it's the other way around?)

the 2nd derivative test is generally easier:
  if you're at a critical point and the -Hessian- is positive, then
  you're a local min; 
  if the Hessian is negative, then you're a local max
  (by positive/negative I mean a ``positive definite'' or ``negative
  definite'' inner product/matrix)
This encourages one to discuss geometry and curvature.

I don't know how to generalize higher derivative tests:
it seems you naturally get a $n^k$
(i.e., $n\times n \times \dots \times n$)
higher-dimensional matrix-thing

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the 2nd derivative test is rarely *used* in intro calc, as you can
virtually always find $f'$ and analyze it explicitly.
If you want students to use it/remember it (and I'm not sure it's so
valuable), you can either:
- give artificial functions, where you only specify the $2$-jet
  ...hence forcing them to apply the 2nd deriv. test
- craft functions whose behaviour at a point is easy to determine, but
  whose local behaviour isn't.
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A dumb mneumonic for the 2nd derivative test/local behaviour is:
+ = smile
- = frown
0 = poker face/can't tell
    (NB: *not* flat: only flat to 2nd deriv: higher terms dominate)
