this is my first question on this community. I am a applied scientist, not a mathematician.

I have the following simplified problem:

Let $u: [0,1] \rightarrow \mathbb{R}_+$ a real valued function and $k\in \mathbb{R}$. The function $u(\cdot)$ is decreasing and may be continuous or not. Let $x^*(k)$ the value that satisfies $$ u\left(x^*(k)\right) - k = u\left(1-x^*(k)\right). $$

I need to get the numerical value $x^*(k)$ for any arbitrary $u(\cdot)$, using a computational Rscript.

Intuitively, I have decided to follow this procedure: I define a deviation $e(x) = u(x) - k - u(1-x)$. Valued at $x^*(k)$, $e\left(x^*(k)\right) = 0$. Then,
  $$ x^*(k) = \arg \min_{x\in [0,1]} e(x)^2. $$

The code is not the problem. I wrote a script that get the graph of function $x^*(k)$ using any decreasing function $u$, i.e. $u=ae^{-bx}$, $u=a - bx^2$, ..., or other discontinuous examples.

I want to known:

1. What is the mathematical/theoretical name of this procedure?
2. In which bibliography can I learn about that?

Thanks.

This is my first question on this community. I am a applied scientist, not a mathematician.

I have the following simplified problem:

Let $u: [0,1] \rightarrow \mathbb{R}_+$ a real valued function and $k\in \mathbb{R}$. The function $u(\cdot)$ is decreasing and may be continuous or not. Let $x^*(k)$ the value that satisfies $$ u\left(x^*(k)\right) = k. $$

I need to get the numerical value $x^*(k)$ for any arbitrary $u(\cdot)$, using a computational Rscript.

Intuitively, I have decided to follow this procedure: I define a deviation $e(x) = k - u(x)$. Valued at $x^*(k)$, $e\left(x^*(k)\right) = 0$. Then, $x^*(k)$ minimizes the squared deviation or the absolute value deviation. $$ x^*(k) = \arg \min_{x\in [0,1]} e(x)^2. $$

The code is not the problem. I wrote a script that return the graph of function $x^*(k)$ using any decreasing function $u$, i.e. $u=ae^{-bx}$, $u=a - bx^2$, ..., or other more complicated examples.

Now, I want to known:

1. What is the mathematical/theoretical name of this procedure?
2. In which bibliography can I learn about that?

Thanks.

this is my first question on this community. I am a applied scientist, not a mathematician.

I have the following simplified problem:

Let $u: \mathbb{R} \rightarrow \mathbb{R}$ a real valued function and $k\in \mathbb{R}$. The function $u(\cdot)$ may be continuous or not. Let $x^*$ the value that satisfies $$ u(x^*) - k = u(1-x^*). $$

I need to get the numerical value of $x^*$ for any arbitrary $u(\cdot)$, using a computational Rscript.

Intuitively, I have decided to follow this procedure: Let $e= u(x) - k - u(1-x)$, then $$ x^* = \arg \min_x e^2. $$

In R there is optimize(...) command that get the minimum, so, the code is not the problem.

I want to known:

1. What is the name of this procedure?
2. In which bibliography can I learn about that?

Thanks.

this is my first question on this community. I am a applied scientist, not a mathematician.

I have the following simplified problem:

Let $u: [0,1] \rightarrow \mathbb{R}_+$ a real valued function and $k\in \mathbb{R}$. The function $u(\cdot)$ is decreasing and may be continuous or not. Let $x^*(k)$ the value that satisfies $$ u\left(x^*(k)\right) - k = u\left(1-x^*(k)\right). $$

I need to get the numerical value $x^*(k)$ for any arbitrary $u(\cdot)$, using a computational Rscript.

Intuitively, I have decided to follow this procedure: I define a deviation $e(x) = u(x) - k - u(1-x)$. Valued at $x^*(k)$, $e\left(x^*(k)\right) = 0$. Then,
$$ x^*(k) = \arg \min_{x\in [0,1]} e(x)^2. $$

The code is not the problem. I wrote a script that get the graph of function $x^*(k)$ using any decreasing function $u$, i.e. $u=ae^{-bx}$, $u=a - bx^2$, ..., or other discontinuous examples.

I want to known:

1. What is the mathematical/theoretical name of this procedure?
2. In which bibliography can I learn about that?

Thanks.