2.6.1—Root location

If f is continuous and f(a) and f(b) have opposite signs, at least one root lies in (a,b). A bracket locates a root; it does not necessarily prove uniqueness.

Check continuity and sign values, then narrow the interval by bisection or another allowed method. State the interval and its width when reporting accuracy.

f(1)<0 and f(2)>0 implies a root in (1,2) for continuous f; repeated bisection gives a guaranteed bracket.

A sign change can hide an odd number of roots, and a zero endpoint must be treated separately rather than called an interior root.