Intermediate secant slope from endpoint derivatives ($f^{\prime }(a)<k<f^{\prime }(b)⟹\frac{f(b^{\prime })-f(a^{\prime })}{b^{\prime }-a^{\prime }}=k$)

Let $m=(f(b)-f(a))/(b-a)$ for a function differentiable on a neighborhood of $[a,b]$. If $m=k$, use the whole interval. If $m>k$, the continuous extension of $(f(x)-f(a))/(x-a)$ takes values $f^{\prime }(a)$ and $m$ at the endpoints, so the intermediate value theorem supplies a secant of slope $k$ starting at $a$. If $m<k$, use the continuous extension of $(f(b)-f(x))/(b-x)$ and obtain a secant ending at $b$. The mean value theorem then produces an interior derivative equal to $k$, without requiring a continuous derivative.