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What should I do when the f(x, y) function's second derivatives or Δ=AC-B² is zero? When the function is f(x) then we can differentiate it until it won't be a zero, but if z = some x and y then can I just continue this process to find what max and min (extremes) it has?

What I've done is:

Differentiated z=f(x, y) by partial derivatives with respect to x and y;

Made them equal to zero in the system of numbers;

Where I got x=0, y=0;

Differentiated function by the second partial derivatives;

Used Δ = AC-B² where if Δ=0 then it needs more calculations.

I got Δ=0.

What should I do now? Differentiate again (third partial) as we do to function f(x) with only one variable x until it won't be equal to zero?

What I've done is:

Differentiated z=f(x, y) by partial derivatives with respect to x and y;

Made them equal to zero in the system of numbers;

Where I got x=0, y=0;

Differentiated function by the second partial derivatives;

Used Δ = AC-B² where if Δ=0 then it needs more calculations.

I got Δ=0.

What should I do now? Differentiate again (third partial) as we do to function f(x) with only one variable x until it won't be equal to zero?