Let $M_m$ be a compact $C^1$ surface in $\mathbb{R^n}$. Prove that there exists $x,y,\in M_m$ such that the distance between them is greatest among all pairs on the surface. Then show that the diameter joining them intersects the surface perpendicular at both points $x$ and $y$.
I think the first part can be shown because the distance function $d(x,y)=||x-y||$ on a compact set has a maximum. I am not sure how to approach the second part of the question.
OK for the first part (and a product of two compact sets is compact). For the second, fix $y$ and take any smooth curve on $M$ passing through $x$ at $t=0$. Express the fact that the distance from $y$ to $x(t)$ has a maximum at $x = x(0)$. Use that to show that the vector from $x$ to $y$ is orthogonal to any vector tangent to the surface at $x$.
Edit: a few more details... Let $x(t)$ be a smooth parameterized curve on M, with $x(0) = x$. For any tangent vector to $M$ at $x$ there is such a curve so that the tangent vector equals $\frac{{\text d} x(t)}{{\text d} t}|_{t=0}$. Now the function $f(t) = \| x(t) - y \|^2$ has a maximum at $t=0$. The derivative of $f$ at $0$ is 2 times the dot product of the vector $\vec {yx}$ and the "velocity vector" of the curve (which is a tangent vector to the surface).