 Residual root error bound

Suppose $f$ is continuous on an interval, differentiable in its interior, and $|f^{\prime }|\ge m>0$ throughout the interior. If $x_{\ast }$ is a root in the interval and $\hat{x}$ is another point in it, the mean value theorem gives $f(\hat{x})-f(x_{\ast })=f^{\prime }(\xi )(\hat{x}-x_{\ast })$ for an intermediate point $\xi$. Taking absolute values proves the stated bound from the residual of a root equation. This converts an observed residual into a stopping certificate, provided the evaluation error in that residual is also controlled.