Consider the following proposition: Let $x_1<x_3<x_2$ be distinct real numbers and $f\in\mathcal{C}^1(\mathbb{R};\mathbb{R})$. If \begin{equation}\max\{f(x_1),f(x_2)\}<f(x_3),\end{equation} then there is a critical point $\xi$ of $f$ in $(x_1,x_2)$ characterized by \begin{equation}f(\xi)=\max\limits_{x\in[x_1,x_2]}f(x)=\inf\limits_{[a,b]\in\Gamma}\max\limits_{x\in[a,b]}f(x),\end{equation} where \begin{equation}\Gamma=\{[a,b]:a\leq x_1<x_2\leq b\}.\end{equation} Question and Remarks: From calculus 1, it clear to me why there is a critical point $\xi\in(x_1,x_2)$ such that $f(\xi)=\max\limits_{x\in[x_1,x_2]}f(x)$. However, I am unable to give a simple interpretation or just picture what is being claimed with the characterization $f(\xi)=\inf\limits_{[a,b]\in\Gamma}\max\limits_{x\in[a,b]}f(x)$.
I am aware that such a proposition is used to illustrate the duality geometry and compactness of the Mountain pass theorem used in nonlinear analysis.
It's simpler than it seems: the relevant observation is that if $[a,b]$ is a superset of $[x_1,x_2]$, then $\mathop{\max}\limits_{x \in [a,b]} f(x) \geq \mathop{\max}\limits_{x \in [x_1,x_2]} f(x)$. Since the maxes over all intervals containing $[a,b]$ are greater, the max over $[x_1,x_2]$ must be the infimum.