When the Hessian=0 and fxx=0

Well, if it's a maximum, what would have to be true about f(x, 0) for small x? (Compared with f(0,0)).
And if it's a minimum, what would have to be true about f(y, 0) for small x? (Compared with f(0,0)).+

not sure?

If you fix one variable, then you're left with a function of just one variable - i.g. f(x,0) becomes g(x)

So, consider what DFranklin is asking in regard to that.

Okay so when x=0 it becomes y^3 which has a pt. of inflection (since the 1st derivatives around small values of 0 don't change signs) and when y=0 it become x^3 which again has a pt of inflection, is that why the function is a saddle point?

..

But wiki seems pretty clear that "if it ain't a min/max, it's a saddle" - they even give a point of inflexion as an example.

Thanks for that. I was using wiki as my reference too. lol.

However, to back it up, I've found a definition in Apostol's Calculus, refering to a point "a".
"A stationary point is called a saddle point if every n-ball B(a) contains points x such that f(x) < f(a) and other points such that f(x) > f(a)".

following that line of reasoning, what if when I fix each variable in turn, I end up with one of them with a max and another with a pt of inflection? IE-- for x^3 - 3xy^2 + y^4.

So z=f(0,y) = y^4 (min. pt)
and z=f(x,0) = x^3 (inflection)

how does that indicate that the function has a saddle pt?

Fixing one of the variables, means that you're just looking at a small set of the values around the point you're interested in.

But there is sufficient information there, in the f(x,0), for instance, to tell you that any neighbourhood of (0,0) is going to contains points that are > f(0,0) and others that are < f(0,0), where f(0,0) is of course 0.

Okay, but what about for a function like this one where the Hessian=0 again: z= x^4 - y^4? (it's supposed to be a saddle pt)

In any neighbourhood of (0,0), there are points greater than 0, and others less than 0, the value of z at (0,0). The same process as before will suffice.

So do we evaluate the entire function for small values around (0,0) or do we evaluate only the functions we get by fixing x=0 and then y=0 in turn?