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I have found that in many optimization problems, there is some sort of "local symmetry" phenomena going on. For example, suppose we have a 'nice' function which has $f'(x)=0$, then we will find that around the points $x_o$ in domain where $f'(x_o)=0$, the function's graph is symmetric about a vertical line passing through $x_o$. Even in higher dimensions, we can see this phenomena, for example, see this recent question posted and it's commentary. Here is another problem where I use a similar idea.

Is there some fundamental mathematical truth going on here?

I have learned group theory and that's suppose to quantify our description of symmetry but I can't seem to make a connection of the ideas taught in it and what's going on here?

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  • $\begingroup$ I think it would depend on what you mean by 'nice' and 'locally symmetric'. This graph has a relative min at zero, is it 'nice'? desmos.com/calculator/d7cme41a5b $\endgroup$
    – David P
    Commented Aug 2, 2022 at 7:23
  • $\begingroup$ There is a sort of local symmetry going on I guess. If you taylor expand it, the quadratic term is leading over leading to symmetry @DavidP $\endgroup$
    – Cathartic Encephalopathy
    Commented Aug 2, 2022 at 8:26
  • $\begingroup$ Asymmetry about a local optimum is quite common in constrained optimization when at least one constraint is active. $\endgroup$
    – Mark L. Stone
    Commented Aug 2, 2022 at 14:55
  • $\begingroup$ Slightly tangential cautionary remark: Despite what we anticipate from experience, sometimes the extreme points of a function with $f(x,y)=f(y,x)$ are not on $x=y$. $\endgroup$
    – Ted Shifrin
    Commented Aug 2, 2022 at 21:51

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