<s>[INST] Find the maximum value of the expression ${\frac {x}{1+x^2}} + {\frac {y}{1+y^2}}+{\frac {z}{1+z^2}}$. 
Find the maximum value of the expression :
  $${\frac {x}{1+x^2}} + {\frac {y}{1+y^2}}+{\frac {z}{1+z^2}}$$
  where $x,y,z$ are real numbers satisfying the condition that $x+y+z=1$.

Taking $x=y=z=\frac {1}{3}$, I get the expression as $\frac {3x}{1+x^2}$, which is equal to $\frac {1}{1+{\frac{1}{9}}}$ or $\frac {9}{10}$.
How can I actually solve the problem without making unnecessary assumptions ?
 [/INST] Threw come to mind.
The first is Lagrange Multipliers.
The second is to reason directly. Plus in $(x,y,z)$ and then plug in $(x-a,y+b,z+c)$ where $a+b+c=0$. You can use algebra to show that the former produces a larger total than the latter, and it follows that the maximum occurs when all the values are equal.
The third is that by symmetry. First observe that the max must have non-negative coordinates. That condition, plus $x+y+z\leq 1$ define a convex optimization problem, whose maximum must be at a corner of the convex hull. It's simple to check that of the corners, $x=y=z$ gives the largest value.
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