If a function f(x,y) has a local or absolute extreme value at the point (x₀, y₀) in its domain, then (x₀, y₀) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

Question

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For a function f(x, y) to have a local or absolute extreme value at a point, that point must either be a critical point (where the gradient of f is zero or undefined), a singular point (where the function or its derivatives are not defined), or a boundary point of the domain of f (where the function can't be further analyzed in all directions due to constraints of the domain).

If a Function F(x,y) Has a Local or Absolute Extreme