# Proving global minimum by lower bound of 2-variable function $f(x,y)=(x-2y)^4+64xy$

I would like to prove that the following function $$f :\mathbb{R}^2\to\mathbb{R}$$ has a global minimum:

$$f(x,y)=(x-2y)^4+64xy$$

I've found the gradient and the hessian of this function, along with it's local minima. I need to prove that those local minima are also global minimums.

$$f$$ has strict local minima at $$f(1,-1/2)=-16$$ and $$f(-1,1/2)=-16$$

I think that what I need to show is that $$-16$$ is a lower bound of this function, and then conclude that's a global minimum, but I didn't manage to do so.

WolframAlpha Link

Please advise. Thank you.

## 1 Answer

Put $$u = x, v = -2y \implies f(u,v) = (u+v)^4 - 32uv$$. Thus you prove: $$f(u,v) \ge -16 \iff (u+v)^4 - 32uv + 16 \ge 0$$. If $$u, v \ge 0$$, then by AM-GM inequality: $$(u+v)^4 \ge \left(2\sqrt{uv}\right)^4 = 16(uv)^2\implies (u+v)^4-32uv+16 \ge 16(uv)^2 - 32uv+16 = 16(uv-1)^2\ge 0$$. Equality occurs when $$uv = 1$$ which corresponds to $$(x,y) = (\pm 1, \mp \frac{1}{2})$$ that was discussed before. If $$u \le 0, v \ge 0$$, then this corresponds to $$x \le 0, y \le 0$$, thus $$xy \ge 0$$, and $$f(x,y) = (x-2y)^4 + 64xy \ge 64xy \ge 0 > -16$$. If $$u \ge 0, v \le 0$$, then $$x \ge 0, y \ge 0\implies f(x,y) \ge 64xy \ge 0 > -16$$. If $$u \le 0, v \le 0$$, then put $$m = -u, n = -v \implies f(m,n) = (-m-n)^4 - 32(-m)(-n)= (m+n)^4-32mn\ge -16$$ by AM-GM again for the pair $$(m,n)$$ with $$m \ge 0, n\ge 0$$. Thus we've shown $$f(x,y) \ge -16$$ for all $$(x,y) \in \mathbb{R^2}$$, confirming $$-16$$ is the global minimum for $$f$$.

• How do you prove that's a global mimimum? That's what I'm trying to prove. Apr 10, 2021 at 17:57
• look here: math.tamu.edu/~tom.vogel/gallery/node16.html I doubt your claim Apr 10, 2021 at 18:02
• Actually, that was the reason why I opened this post. I didn't manage to figure how to do it :( Apr 10, 2021 at 18:10
• Yes I know that. I would like to prove it algebraically. Could you help me to do so? Apr 10, 2021 at 18:41
• @Dennis: I redo it. Please take a look at the edited work. Let me know if I miss any thing.....
– user899577
Apr 10, 2021 at 20:49