Edit: I feel this question is better suited for MathOverflow, so I added a significant amount of context and posted there. I don't know if I should delete the question on this site, but I feel I should provide the link to not duplicate work. Here is the link: https://mathoverflow.net/questions/375682/how-can-i-analyze-the-the-affect-of-a-constant-on-the-arguments-that-minimize-a
Consider the a function $f(x,y,c)$ where $c$ is a constant. I just wrote $c$ as an argument to emphasize the function contains an important parameter $c$. Now, consider
$$\text{argmin}_{x,y} \; f(x,y,c)$$
How can I either (1) show that the arguments that minimize $f$ is not affected by the choice of $c$, or furthermore, (2) how can I determine how much the parameter $c$ affects the arguments that minimize $f$. I am mostly interested in (2), but I am convinced I need to understand (1) to work out (2).
While the problems I am typically working with are not so simple, I will provide a simple example to illustrate my question.
Example 1: Consider $$ f(x,y,c) = cx^2 + y^2 $$ where $c > 0$. The minimum of this function is $x=0$ and $y=0$ regardless of the value of $c$. This is simple to see for this function.
Example 2: Consider the following equations, $$ f_1(x,y,c) = (x+c*0.00001)^2 + y^2 $$ $$ f_2(x,y,c) = (x+c)^2 + y^2 $$
The value of $c$ in the first equation $f_1$ does not affect the minimizing arguments as much as the value of $c$ in the second equation $f_2$.
This answer is sourced from Oregon State University Calculus Undergraduate Study Guide. [See References]
Compute $f'_x(x,y,c)$ and $f'_y(x,y,c)$ the partial derivatives of $f(x,y,c)$ w.r.t. $x$ and $y$ respectively.
Equate the partial derivatives to $0$ and solve them simultaneously to determine critical points (i.e., maxima or minima) where the tangent plane is horizontal with its normal pointing in the $z$ direction.
Let $(x_c,y_c)$ be a critical point on the 2D surface given by $f(x,y,c)$.
Define $D(x_c,y_c) = f''_x(x,y,c)f''_y(x,y,c) - [f''_{xy}(x_c, y_c, c)]^2$
where
$f''_x(x,y,c)$ and $f''_y(x,y,c)$ are the second derivatives w.r.t $x$ and $y$ respectively and
$f''_{xy}(x,y,c)$ is the second derivative of $f(x,y,c)$ w.r.t $x$ first followed by $y$ next.
We have four cases:
If $D>0$ and $f_xx(x_c,y_c)<0$, then $f(x,y)$ has a relative maximum at $(x_c,y_c)$.
If $D>0$ and $f_xx(x_c,y_c)>0$, then $f(x,y)$ has a relative minimum at $(x_c,y_c)$.
If $D<0$, then $f(x,y)$ has a saddle point at $(x_c,y_c)$.
If $D=0$, the second derivative test is inconclusive.
From this we can see that for the question posed, if $c$ appears in the partial derivatives $f'_x(x,y,c)$ or $f'_y(x,y,c)$ we can conclude that it impacts the maxima or minima for general $f(x,y,c)$. The extent of the impact is defined exactly by the system of two equations that equate the partial derivatives to $0$ in the computation of the critical points.
References:
Maxima and Minima of Functions of Two Variables. Oregon State University, Calculus study guide.