Taylor's theorem-multivariable

Question

We are given a function: $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a $C^{2}$ function such that:

$D^{2}f\left ( x \right )\left ( t,t \right )> 0$, $\forall x\in \mathbb{R}^{n}, \forall t\in \mathbb{R}^{n}-\left \{ 0 \right \}$

1- Show that: $f\left ( y \right )> f\left ( x \right )+Df\left ( x \right )\left ( y-x \right )$ for every $x,y\in \mathbb{R}^{n}, y\neq x$.

2- Deduce from the previous part that $Df\left ( x \right )$ cannot vanish at more than one point $x$.

For the first part, the inequality follows immediately from the Taylor's theorem and the fact that $D^{2}f\left ( x \right )\left ( t,t \right )> 0$.

Can you please help me with the second part? I have no idea how to solve it.

Answer 1

If $Df$ vanished at distinct points $x,y$, we would see $f(y) > f(x)$ and $f(x) > f(y)$ by applying the first part to $x$ and $y$ successively. Problem.