Kinematic meaning of this mean value theorem?

Question

Here is another version of the mean value theorem :

Let $f,g,h$ three real functions continue on $[a,b]$ and differentiable on $]a,b[$.

We define $\phi(x) = \begin{vmatrix} f(a) & f(b) & f(x)\\ g(a) & g(b) & g(x)\\ h(a) & h(b) & h(x) \end{vmatrix}$

and show that there exists $c\in ]a,b[$ such that $\phi'(c)=0$.

But I was wondering what was the kinematic meaning of this theorem for the following motion : $t \mapsto M(t)=(f(t),g(t),h(t))$ ?

Is it possible to have a drawing of the situation ?

The nullity of the determinant means that the three vectors are in the same plane.

Thanks in advance !

Answer 1

Slightly reinterpreting my previous comment, for any $c$ with $\phi'(c)=0$, $M(c)$ will give you a point on the curve $\{M(t)\}$ where the height of the curve over the plane spanned by $M(a)$ and $M(b)$ has a critical point. In particular, one of those critical points gives the point of maximal distance from that plane.