Let scalar field $f : \mathbb{R}^n \to \mathbb{R}$ be smooth, i.e., in $C^\infty (\mathbb{R}^n)$, and there exists $\bar{x}$ be a point on the line segment connecting $x_1$ and $x_2$. Is the following gradient inequality correct?
$$|f(x_1)-f(x_2)| \leq | \nabla f(\bar{x})| \cdot |x_1 -x_2| $$
where $| \cdot |$ denotes the Euclidean norm.
Using the gradient theorem: \begin{align} \left|f\left(x_2\right) - f\left(x_1\right)\right| &= \left|\int_{0}^1 \left\langle \nabla f\left((1-t)x_1 + tx_2\right), x_2 - x_1\right\rangle \mathrm dt\right|\\ &\le \sup_{t\in [0,1]} \left\|\nabla f\left((1-t)x_1 + tx_2\right)\right\|\left\|x_1-x_2\right\| \end{align}
Since $t\mapsto \left\|\nabla f\left((1-t)x_1 + tx_2\right)\right\|$ is a continuous function and $[0,1]$ is compact, you can have $t_0\in [0,1]$ such that $\left\|\nabla f\left((1-t_0)x_1 + t_0x_2\right)\right\| = \sup\limits_{t\in [0,1]} \left\|\nabla f\left((1-t)x_1 + tx_2\right)\right\|$. The statement you are looking for is obtained by having $\overline x = (1-t_0)x_1 + t_0x_2$