Formulas: max(f,g) = (f + g + |f - g|) / 2 min(f,g) = (f + g - |f - g|) / 2

Differentiability: max(f,g) and min(f,g) are non-differentiable where f(x) = g(x) AND f'(x) != g'(x) (the curves cross with different slopes).

Example: max(x, $x^2$) on R. x = $x^2$ at x = 0 and x = 1. At x = 0: derivatives are 1 and 0 (different) — not differentiable. At x = 1: derivatives are 1 and 2 (different) — not differentiable.

If f(x) = g(x) and f'(x) = g'(x): The max/min IS differentiable (smooth transition).