<s>[INST] Minimum of $\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}$ for $0\leq x+y,y+z,z+x\leq 1$ Let $0\leq x,y,z,x+y,y+z,z+x\leq 1$. What is the minimum of $$F(x,y,z)=\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}?$$
We have $F(1/2,1/2,1/2)=4/3$. Since the constraints are on $x+y,y+z,z+x$ instead of $x,y,z$, taking partial derivative with respect to $x,y,z$ doesn't help much. We cannot change one variable without affecting the others.
 [/INST] Hint: Do the substitution $ a= x+y, b = y + z, c = z + x $, and we have $ 0 \leq a, b, c, \leq 1 $ and we want to minimize
$$ \frac{1}{a} + \frac{1}{c}  - \frac{2}{a+b+c}. $$
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