Bordered Hessian matrix to find a minimum of the function

Question

I was trying to find the global minimum for the function $$(a + b) z + (a + c) y + (b + c) x $$ subject to the following constraint: $$(xy + xz + yz)(ab + bc + ac)=1.$$ By Lagrange multipliers I found $2$ as a critical value, but calculating the bordered Hessian I did not find the answer. Could anyone there help me, please?

Answer 1

For real variables the minimum does not exist.

For non-negative variables by C-S twice we obtain: $$(a+b)z+(a+c)y+(b+c)x=(x+y+z)(a+b+c)-(ax+by+cz)=$$ $$=\sqrt{(x^2+y^2+z^2+2(xy+xz+yz))(a^2+b^2+c^2+2(ab+ac+bc))}-(ax+by+cz)\geq$$ $$\geq\sqrt{(x^2+y^2+z^2)(a^2+b^2+c^2)}+2\sqrt{(xy+xz+yz)(ab+ac+bc)}-(ax+by+cz)\geq$$ $$\geq ax+by+cz+2\sqrt{(xy+xz+yz)(ab+ac+bc)}-(ax+by+cz)=2.$$ The equality occurs for $a=b=c=x=y=z=\frac{1}{\sqrt3},$ which says that we got a minimal value.