I am reviewing masters exams and can't recall the multivariable calculus one needs to prove that this is true. A reference would suffice. Thank you!
Suppose $x_1,x_2,x_3 \in \mathbb{R}^2$ are three points in the plane that do not lie on a line, and denote by $T$ the closed triangle with vertices $x_1,x_2,x_3$. Suppose $f : T \to \mathbb{R}$ is a continuous function which is differentiable on the interior of $T$ and which vanishes on the boundary of $T$.
Prove that there exists a point $x$ in the interior of $T$ such that the tangent plane to the graph of $f$ at the point $x$ is horizontal.
EDIT: I've been looking through the Wikipedia links, and I understand that for every line there is a point where the derivative must be zero, but I don't see how to the prove the existence of a point where the derivative along every line must be zero. So I may need to be walked through this a little more, if you have time.