Is symmetry a valid option in inequalities?

Question

Consider two questions:

Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal since the equation is symmetrical in all other variables so, then $a=b=c=d=k$ (let) which gives $e=16/5$, neglecting other roots.

Q2. $$\frac8x+\frac1y=1$$ Minimize $x+y+\sqrt{x^2+y^2}$. My answer: similiarly to previous problem now second equation is symetrical in x and y so when $x=y=9$, minimum value of expression is $18+9\sqrt2$

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I want to ask, is my approch right?Can inequalities be dealt like this?Is symmetry a valid option?

Answer 1

The reasoning of symmetry applies on (1) but not (2), as some of the comments have noted.

In (1), replacing $a, b, c, d, e$ with any permutation of them would not change the question at all. With questions similar to (1) it is quite often to state:

Without loss of generality, assume $a \le b \le c \le d \le e$, then we have blah blah blah.....

EDIT: My answer doesn't entirely cover the original question (1), which also deduces $a=b=c=d$ if $e$ attains maximum value, which is wrong. That extra condition is usually an assumption from solver encountering such symmetric equations, but never explicitly or implicitly implied in original question.

In (2), exchanging $x$ and $y$ immediately invalidates $\frac8x+\frac1y=1$, except for the very specific case $x=y$, which is an additional constraint not in the question itself.

I think this is what the OP wants: whether the reasoning of symmetry of variables apply to specific questions, instead of solving them here.

Answer 2

Consider the following problem: Maximize $x^2+y^2$ under the constraint $|x|+|y|=1$. Unless I am mistaken, you will not find your suggested method succesful.

Answer 3

For question 1 You are right that $a = b= c= d $ and letting that be equal to $k$ is the right way to think. But, you made $k=e$, which is not what we want. We want to minimize k and maximize x.

I made the two equations $4k+e=8$ and $4k^2+e^2=16$ and solved for $e$ in terms of $k$ for one, plugged it into the other. It gave a quadratic equation with the higher $e$ being your maximized function.

Trying out question 2, will edit if I get something.