Finite dimensional "Mountain Pass Lemma" Question Does anyone know of a good reference which I can cite for the finite dimensional version of Mountain Pass Lemma?
Motivation I am writing a paper and found myself using the following result:

Let $f$ be a proper smooth real-valued function on $\mathbf{R}^3$ such that $f(0) = 0$, $f|_{B_1(0)} \geq 0$, $f|_{\partial B_1(0)} \geq 1$ and $\exists p \in {\partial B_2(0)}$ such that $f(p) = 0$. Then $\exists q\in \mathbf{R}^3 \setminus B_1(0)$ such that $f'(q) = 0$ and $f(q) \geq 1$. 

For the time being I referred to Ambrosetti and Rabinowitz's JFA article for the mountain pass lemma, but citing a Banach space version for a finite-dimensional Euclidean space application gives me a funny feeling. (Also, if feels like such a result could in principle be found in not-so-advanced undergraduate textbooks...)
 A: For historical interest: A friend pointed me to the book 


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*Youssef Jabri, The Mountain Pass Theorem: Variants, Generalizations and Some Applications, CUP


which asserts that one of the earliest known published version of the finite dimensional mountain pass theorem was due to 


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*Richard Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience


published originally in 1950. The version stated and proven by Courant does not, technically speaking, imply the result I stated in the question text (the points $0$ and $p$ are assumed to be local minima of the function $f$). But a simple modification of the deformation lemma (for example, as in Liviu's book that he mentioned) would do. 
A: I have stumbled across Richard Palais' (co-author Chuu-lian Terng) Critical Point Theory and Submanifold Geometry (Springer Lecture Notes in Math 1353). This is an awesome book!
The "Mountain Pass Lemma" for finite-dimensional manifolds is presented as Theorem 9.2.7 (pg189).
A: This seems to be in L Evans's PDE book, section 8.5
A: My book An Invitation to Morse Theory,   2nd Edition,   Springer Verlag 2011    describes the  finite dimensional Mountain Pass Lemma in Example  2.53.  There I work  on a compact manifold, but the compactness of the  manifold can be substituted by a properness assumption on the function. In the same  section I explain a more general Min-Max principle (Thm. 2.51) and in Example 2.53 I explain  how this implies the Mountain Pass Lemma.
