For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowlegde to solve it.

**Problem definition:** Let $f(\xi) \in \mathcal{C}^1$ be a smooth function with mapping $f:\mathbb{R}^n \to \mathbb{R}^p$ with $p\le n$. With this information, we have the term:
\begin{equation}
\left(f(\xi)-f(\tilde{\xi})\right)^\top\left(\frac{\partial f}{\partial \xi}\left(\bar{\xi}\right)\right)\left(\xi-\tilde{\xi}\right) \qquad\qquad\text(1)
\end{equation}
Where $\xi, \bar{\xi}, \tilde{\xi} \in \mathbb{R}^n$ and varying over time. Using the mean-value-theorem, we can prove that
$$\left(f(\xi)-f(\tilde{\xi})\right)^\top\left(\frac{\partial f}{\partial \xi}\left(\bar{\xi}\right)\right)\left(\xi-\tilde{\xi}\right)\le \left(\xi-\tilde{\xi}\right)^\top \left(\frac{\partial f}{\partial \xi}\left(\bar{\xi}\right)\right)^\top\left(\frac{\partial f}{\partial \xi}\left(\bar{\xi}\right)\right) \left(\xi-\tilde{\xi}\right)$$
Thus, for $(1)$ we know that we can find an upperbound for all $\xi, \bar{\xi}, \tilde{\xi} \in \mathbb{R}^n$. However, for the problem, we require a lower bound for all $\xi, \bar{\xi}, \tilde{\xi} \in \mathbb{R}^n$ or show that the $\le$ sign can be changed to an equality sign for all $\xi, \bar{\xi}, \tilde{\xi} \in \mathbb{R}^n$. However, I do not know how to show it.

Could one of you help me out?