# Is symmetry a valid option in inequalities?

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 $18+9\sqrt2$

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

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Adopting a strategy which yields the same answer when one is asked to maximize (Q1) or to minimize (Q2) a quantity, is suspicious in full generality, don't you think? – Did Jul 31 '14 at 15:31
In Q2, the constraint is not symmetric w.r.t. x and y, so there's no reason (that I can see) to expect the minimum to occur at a point where x = y. – Ned Jul 31 '14 at 15:39
In the second problem, the minimum is not $18+9\sqrt{2}$. – André Nicolas Jul 31 '14 at 15:39
@AndréNicolas Its 26? – ADG Jul 31 '14 at 15:41
Symmetry in the question statement means your solution set is symmetric, individual solutions need not be. So this approach could mislead. For e.g. check maa.org/programs/maa-awards/writing-awards/… – Macavity Jul 31 '14 at 16:54

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.

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So I infer "symmetry is not applicable everywhere in inequalitites" – ADG Jul 31 '14 at 16:49
Yep, it only works when you shuffle all variables and the equation / inequality is still all the same. – Abel Cheung Jul 31 '14 at 16:54
@abelcheung Not quite. Symmetry in the question statement means your solution set is symmetric, individual solutions need not be. So this approach could mislead. For e.g. check maa.org/programs/maa-awards/writing-awards/… – Macavity Jul 31 '14 at 16:56
@Macavity Sorry if the statement looks confusing; what I mean is, when you look at the question itself, the variables are symmetric. Nowhere did I mention the solutions. – Abel Cheung Jul 31 '14 at 16:58
@AbelCheung yes. However that does not mean at the extremum variables should be equal, at best it is an initial guess which needs further proof. E.g find the maximum of $$(x^2+(y-1)^2)((x-1)^2+y^2)$$ where the question is clearly symmetric. – Macavity Jul 31 '14 at 17:06

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.

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yep it give when $|x|=|y|=1/2 \implies (x^2+y^2)_{max}=1/2$ but when $|x|=1,y=0$ or the reverse $(x^2+y^2)_{max}=1$ – ADG Jul 31 '14 at 16:19

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.

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No! you are mistaken I don't want a solution(Actually I want to try on my own). Just tell about symmetry – ADG Jul 31 '14 at 16:32
I gave no solution, only a means to it for question 1. Question 2 fundamentally does not have the same symmetry, unless there is an assumption that $x,y>0$ then one of the two can be negative which would pull down $x+y+\sqrt{x^2+y^2}$, which variable is negative depends on the constraint function. I honestly think the best method to solve this would be using Lagrange multipliers if you have seen them. Otherwise I would tackle the problem by solving one in terms of another then plugging into the function to minimize and setting the derivative equal to $0$. – kleineg Jul 31 '14 at 16:41
anyways thanks for your help – ADG Jul 31 '14 at 16:46