Discussion
If a monotonic and continuous function \(y = f(x)\) has only one root in the interval \(x_{1} \lt x \lt x_{2}\), then
Answer: C
The condition is \(f(x_{1})f(x_{2}) \lt 0\).
A continuous monotonic function with exactly one root strictly between \(x_{1}\) and \(x_{2}\) crosses zero once there, so it has one sign at \(x_{1}\) and the opposite sign at \(x_{2}\). The product of two numbers of opposite sign is negative. This is the condition used to bracket a root in the bisection and false-position methods.
A positive product would mean the same sign at both ends, so a monotonic function would have no root between them. A zero product would put the root at an end point, not inside the interval. \(f(x_{1}) = f(x_{2})\) is impossible for a strictly monotonic function on distinct points.
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