Without using calculus I was told to find the maximum of $$f(x)=6x-x^2.$$
The "symmetry" approach notices that $f(x)=x(6-x)$, and that replacing $x\leftrightarrow6-x$ does not change $f(x)$ which means it doesn't change the maximum. Then the solution says that the only value of $x$ that is unchanged by $x\leftrightarrow 6-x $ is $x=3$. So that's the location of the maximum.
This is all the solution says. I am under the impression that if $f(x)$ is unchanged then no number should be affected unchanged. How did they figure out that $x=3$ is the only thing unchanged?