Existence of global minimum $f(x,y,z) = x + y + z$ under the constraint $x^2+xy+2y^2-z=1$

Question

The full exercise consists of (i) finding the minimum value of $f(x,y,z) = x + y + z$ under the constraint $g(x,y,z)=x^2+xy+2y^2-z=1$, and (ii) establishing whether the function has a maximum.

I have no problems with showing that $f$ has no maxmimum by setting $y = 0$, and showing that for an arbitrary point $x=a$ we can find $z=a^2 -1 $ which satisfies the constraint. I am also able to solve the minimization problem by setting up the Lagrange function and solving it in the usual manner. This produces the point $(-\frac{3}{7},-\frac{1}{7},-\frac{5}{7})$. However, I'm not sure how I can motivate that this truly is the minimum value of $f$.

I suspect that I am supposed to show that $g(\cdot)$ is bounded downwards, but I'm not sure how to go about it.

Any help would be much appreciated!

Answer 1

Hint: with $$z=x^2+xy+2y^2-1$$ we get $$f(x,y,x^2+xy+2y^2-1)=x+y+x^2+xy+2y^2-1$$ and you get a Problem only in two variables

Answer 2

Hint, $g(x,y,z) = 1$ can be rewritten as

$$(x+y)^2 + y^2 = z+1$$

Thus $z+1$ is the sum of two squares, so $z \geq -1$