Given three points in $\mathbb{R}^2$, $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$, and a $C^2$ surface through them $f(x,y)$, is there always a point in the interior of the triangle in formed by the three points where the tangent plane is parallel to the plane through $(x_i,y_i,f(x_i,y_i))$.
I've not seen this as a generalisation of the MVT, so I'm assuming it should be either blindingly obvious, or there should be some simple counter example - but I can't think of either myself.