My intuition behind the generalized mean value theorem is similar to that elaborated in this post. For example, if you have two runners running for 10 seconds, at some point $c$ the ratio of the instantaneous speeds $f'(c)/g'(c)$ is going to be the ratio of the average speeds $\frac{\frac{f(b)-f(a)}{b-a}}{\frac{g(b)-g(a)}{b-a}}$.
However, in Abbott, I am asked to give a geometric interpretation of the generalized mean value theorem by considering $f(x)$ and $g(x)$ as parametric equations of a curve. How does one think about GMVP in this manner?
