optimization with strict inequality of variables

Question

Maximize $f(x_1,x_2, x_3) = x_{2}+x_{3} - (x_{2}^2+x_{3}^2)$ given $\sum_{i=1}^{3}x_{i} = 1$ and $x_{i}>0$ for $i=1,2,3$. I f I assume that $x_{i}\geq0$ for $i=1,2,3$ then the solution is $x_2 = x_3 = 1/2, x_1 = 0$. How to get the solution when we have strict inequlity $x_{i}>0$ for $i=1,2,3$ ?

Answer 1

By C-S and AM-GM $$x_2+x_3-(x_2^2+x_3^2)=x_2+x_3-\frac{1}{2}(1^2+1^2)(x_2^2+x_3^2)\leq x_2+x_3-\frac{(x_2+x_3)^2}{2}=$$ $$=\frac{1}{2}\cdot(x_2+x_3)(2-x_2-x_3)\leq\frac{1}{2}\left(\frac{x_2+x_3+2-x_2-x_3}{2}\right)^2=\frac{1}{2}.$$

The equality occurs for $x_2=x_3$ and when $x_2+x_3=2-x_2-x_3,$

which gives $x_2=x_3=\frac{1}{2}$ and $x_1=0.$

If $x_1>0$ then the maximum does not exist, but $\sup{f}=\frac{1}{2}.$