# Proving global minimum by lower bound of 2-variable functions

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

1. $$f_1(x,y)=x^4+2x^2y+y^2-4x^2-8x-8y=(x^2+y)^2-4(x^2+2x+2y)$$
2. $$f_2(x,y)=(x-2y)^4+64xy$$

I've found the gradient and the hessian of both functions, along with their local minima. I need to prove that those local minima are also global minimums.

• $$f_1$$ has strict local minimum at $$f_1(1,3)=-20$$
• $$f_2$$ has strict local minima at $$f_2(1,-1/2)=-16$$ and $$f_2(-1,1/2)=-16$$

I think that what I need to do is to show that those minimum points are also their lower bounds, but I couldn't prove that. Another idea is to show that they (maybe) coercive functions.

Thank you.

Write $$t = x^2+y$$, then $$y= t-x^2$$ so we have $$f(x,y)=(x^2+y)^2-4(x^2+2x+2y)$$
$$= t^2 -4(x^2+2x+2t-2x^2)$$
$$=t^2-8t +\color{red}{16}+ 4x^2-8x +\color{red}{4} -20$$
$$= (t-4)^2+4(x-1)^2-20$$
$$\geq -20$$
It achieves minimum iff $$x=1$$ and $$t=4$$ so $$y=3$$.