Corollary 2 — On the Sign of the Derivative
Under the conditions of the theorem, the derivative of [latex]f[/latex] is either positive
throughout the interior of [latex]I[/latex] or negative throughout the interior of [latex]I[/latex].
throughout the interior of [latex]I[/latex] or negative throughout the interior of [latex]I[/latex].
Proof
By assumption, the derivative exists and does not take the value 0 in the interior of [latex]I[/latex]. The function is strictly monotone over [latex]I[/latex]. If it is increasing, the derivative cannot be
negative, so it is everywhere positive in the interior of [latex]I[/latex]. If the function is decreasing, the derivative cannot be positive, so it is everywhere negative in the interior of [latex]I[/latex].
