An interesting point regarding critical points and extrema

[coquelicot]

TL;DR Summary

critical points of a function on "critical lines"

Hi all,

I have recently faced some problem about distances between two curves, and (re?)"discovered" an interesting point that I would like to share with you.
In the following, we consider a function of two variables $f(x,y)$, but it should be clear that the definitions and the result is valid for functions of more variables.

First, let me set few definitions:
Let $f(x,y)$ be a differentiable function and $(x_0, y_0)$ be a point in its domain. A critical partial line of f in the neighborhood of $(x_0, y_0)$, with respect to the variable $y$, is a function $\varphi(x)$ defined in the neighborhood of $x_0$ such that $y_0 = \varphi(x_0)$ and $$ {\partial f \over \partial y} (x, \varphi(x)) = 0.$$
Similarly, a critical partial line of f in the neighborhood of $(x_0, y_0)$ with respect to the variable $x$, is a function $\psi(y)$ defined in the neighborhood of $y_0$ such that $x_0 = \psi(y_0)$ and $$ {\partial f \over \partial x} (\psi(y), y) = 0.$$

If furthermore the point $(x_0, y_0)$ is such that
$$ {df (x, \varphi(x))\over dx}\bigg|_{x=x_0} = 0$$
(resp. ${df(\psi(y), y)\over dy}\bigg|_{y = y_0} = 0$), then we say that $(x_0, y_0)$ is a critical point of $f$ on the critical partial line $\varphi$ (resp $\psi$).
With these definitions, we have the following theorem:

The critical points of $f$ on its critical partial lines are the critical points of $f$. In particular, they do not depend on the variable with respect to which the critical partial line refers to.

Before I give the (very simple proof), let me give an application:

Let $y = f(x)$ and $y = g(x)$ be two curves defined in open intervals. Assume that $P_1 = (a_1, f(a_1))$ is a point of curve $f$ that minimize (resp. maximize) the distance $d((x_1, f(x_1), g)$, and $P_2= (a_2, g(a_2))$ is a point of $g$ that minimize (resp. maximize) the distance $d((x_2, g(x_2)), f)$.
Then $(a_1, a_2)$ is a critical point of the distance function
$$h(x_1, x_2) = (x_1 - x_2)^2 + (f(x_1) - g(x_2))^2. $$
Furthermore, the straight line between $P_1$ and $P_2$ is orthogonal to both curves (this is the not so easy point).

Proof: The distance of a point $(a_1, f(a_1))$ to curve $g$ is given by
$$\min_{x_2} h(a_1, x_2).$$ A point of minimum $a_2$ exists and must fulfill
$${\partial h (a_1, a_2)\over \partial x_2} = 0.$$
Moreover, the straight line from $(a_1, f(a_1))$ to $(a_2, g(a_2))$
is orthogonal to curve $g$, as is well known.
By the implicit function theorem, there is a function $\varphi(x_1)$ defined in the neighborhood of $a_1$ such that
$${\partial h (x_1, \varphi(x_1))\over \partial x_2} = 0.$$
So $\varphi$ is a critical partial line of $f$ with respect to $x_2$.
Thus, the problem reduces to find a point $a_1$ that is a critical point of $h$ on $\varphi$.
By the theorem, $(a_1, a_2= \varphi(a_1))$ must be a critical point of $f$.
Furthermore, we have seen that the straight line $(a_1, f(a_1))-(a_2, g(a_2))$ is orthogonal to $g$. By symmetry (which is not evident a priori), taking a critical partial line of $f$ with respect to $x_1$, it is also orthogonal to $f$ (q.e.d).

Proof of the theorem:
Assume for example that $\varphi$ is a critical partial line of $f$ with respect to $y$ at $(x_0, y_0)$.
We have
$${df (x, \varphi(x))\over dx}\bigg|_{x=x_0} = {\partial f(x_0, \varphi(x_0))\over \partial x} + \varphi'(x_0) {\partial f(x_0, \varphi(x_0))\over \partial y}. $$
Hence the condition
$${\partial f(x_0, y_0)\over \partial x} = 0 \ {\rm and} \ {\partial f(x_0, y_0)\over \partial y} = 0$$
is equivalent to the condition
$${df (x, \varphi(x))\over dx}\bigg|_{x=x_0} = 0 \ {\rm and} \ {\partial f(x_0, \varphi(x_0))\over \partial y} = 0 ,$$
but from the hypothese,
$$ {\partial f(x_0, \varphi(x_0))\over \partial y} = 0 $$
since $\varphi$ is a critical line of $f$ at $(x_0, y_0)$, with respect to $y$ (q.e.d.)

 

[jvicens]

Hi there,

Thank you for sharing your interesting discovery with us! It's always exciting to come across new insights and theorems in our field of study.

Your theorem about critical points on critical partial lines seems like a useful tool for analyzing functions of multiple variables. It's fascinating how the critical points on these lines are also critical points of the overall function, and that they are independent of the variable with respect to which the critical partial line is defined. I can see how this can be applied to various problems, such as the one you mentioned about finding the minimum and maximum distances between two curves.

Your proof is clear and well-structured, and it's great that you included an application of your theorem to further illustrate its usefulness. The fact that the straight line between the critical points of the distance function is orthogonal to both curves is indeed not obvious, but your explanation makes it much more understandable.

Overall, I think your theorem and proof are valuable contributions to the field of mathematics, and I appreciate you sharing it with us. I'm sure it will be helpful for other scientists and mathematicians facing similar problems.

Thank you again for your contribution. I look forward to reading more of your work in the future.