What are the properties of a function for which the function attains its maxima or minima when all the variables of the function are equal?

Question

I have noticed that in many functions with one or more variables we can get the maxima or minima if we assume all the variables to be equal. Are there certain properties a function must satisfy for this to be true, or is this completely random?

Take a look at these two functions.

1. Assume ABC is an acute angled triangle and $\displaystyle p=\frac{\sqrt3+\sin A+\sin B+\sin C}{2\sin A\sin B\sin C}$ .If we assume $A=B=C=60^\circ$ we can get the minimum value of $p$.

2. Similarly $y=\sum_{cyc (a,b,c)}{}\frac{a}{b^2+1} \geq \frac{3}{2}$ where $a+b+c=3$ .If we assume $a=b=c$, we can get the minimum value of $y$

There are countless other examples like this, is there anything going on here?

Answer 1

I think it's completely random in the general.

If we want to use AM-GM, we know that the equality occurs for equality case of any variables.

For example, for non-negatives $a$ and $b$ we have $\frac{a+b}{2}\geq\sqrt{ab}$,

which says that a minimal value of $\frac{a+b}{2}-\sqrt{ab}$ is equal to $0$

and the maximal value of $\sqrt{ab}-\frac{a+b}{2}$ is equal to $0$ and this maximum occurs for $a=b$.

Now, let we need to find a maximum of $$(a^2-ab+b^2)(a^2-ac+c^2)(b^2-bc+c^2)$$ for non-negatives $a$, $b$ and $c$ such that $a+b+c=3$.

For $a=b=c=1$ we get a value $3$. But it's not a maximal value.

The maximal value it's $12$ and occurs for $(a,b,c)=(2,1,0)$.