What are the maximum and minimum resultant magnitude of two forces

Let F₁ and F₂ denote the two forces, so the resultant is F₁ + F₂. Then by the triangle inequality,

|F₁ + F₂| ≤ |F₁| + |F₂|

Magnitude must be non-negative, so the minimum magnitude of the resultant is 0. This happens when F₁ and F₂ have the same magnitude, but point in opposite directions.

The maximum magnitude is the sum of the magnitudes of F₁ and F₂. This happens when the magnitudes of F₁ and F₂ are not equal, and the two forces point in the same direction.