Difference between minimum-sufficient and necessary in math.

Question

I'm having some trouble wrapping my head around this concept for some reason.

Take 3 hypothetical questions:

1. "Find the minimal sufficient conditions on X such that Y is Z"
2. "Find the necessary conditions on X such that Y is Z"
3. "Find the conditions on X such that Y is Z"

What are the differences in what these questions are asking?

To put these hypotheticals in a more concrete setting, we can use the following example related to fixed point iteration:

• X: $g(x)$
• Y: $x_{n+1}=x_n+f(x)g(x)$
• Z: "Is at least Quadratically Convergent"