As the title states I would like to find a $g(x)$ such that $f(x+g(x))|_{x=a}$ is maximum/minimum where $f(x)$ is known.
I have looked at Euler-Lagrange equations but they are derived for integral functionals and I am not sure whether it is applicable in my case. (I am also not sure whether such $g(x)$ exists.) Can you please state a solution or provide references for the solution?
I would like to generalize my solution to n dimensional space, thus a solution in one dimensional is greatly appreciated.
Regards,
Your question suggests that you're trying to find $g(x)$ for which $f(x+g(x))|_{x=a}$ is extremal , so you're trying to find extremum in $x=a$ .
In that case you can just differentiate :
$\frac{d}{dx}f(x+g(x))=0 \implies f'(a+g(a))(1+g'(a))=0$. This doesn't give you a requirement for a function $g()$ but rather two numbers : $g(a)$ and $g'(a)$. You can choose $g()$ such that $a+g(a)$ gives a zero value for $f'()$, or such that $g'(a)=-1$. So it only gives you a requirement for $g()$ and / or $g'()$ in point $a$. Rest is free to choose.
PS : Euler-Lagrange and variational calculus are usually used to find extremum of an integral e.g. :
$\int_{a}^{b} f(x+g(x)) dx $ .
In such cases the outcome will be a differential equation with which you can find functions $g()$ that give extrema for the integral. From the above you can see why this is not applicable or necessary in your case.