We are given a function:
\[ f(x) = x^2 + ax + b \]
Define a new function:
\[ g(x) = f(x+1) - f(x-1) \]
\[ f(x+1) = (x+1)^2 + a(x+1) + b = x^2 + 2x + 1 + ax + a + b \] \[ f(x-1) = (x-1)^2 + a(x-1) + b = x^2 - 2x + 1 + ax - a + b \]
\[ g(x) = f(x+1) - f(x-1) \] \[ = [x^2 + 2x + 1 + ax + a + b] - [x^2 - 2x + 1 + ax - a + b] \] \[ = 4x + 2a \]
\[ g(20) = 4(20) + 2a = 80 + 2a = 72 \Rightarrow 2a = -8 \Rightarrow a = -4 \]
\[ f(x) = x^2 - 4x + b = (x - 2)^2 + (b - 4) \]
Since a square term is always non-negative, the minimum value of \( f(x) \) is at \( x = 2 \), and that minimum is: \[ f(2) = (2 - 2)^2 + (b - 4) = b - 4 \] To ensure \( f(x) \ge 0 \), we need: \[ b - 4 \ge 0 \Rightarrow b \ge 4 \]
The minimum value of \( b \) is \( \boxed{4} \).