Y by
Consider the function 
defined on 
by
![\[f(x, y) = x^4 + y^4 - 2(x - y)^2.\]](//latex.artofproblemsolving.com/1/d/9/1d9e85722971f67e3ba2dee0e12d119ceabc671a.png)
Show that there exist
(and determine them) such that
![\[\forall (x, y) \in \mathbb{R}^2, f(x, y) \geq \alpha \| (x, y) \|^2 + \beta,\]](//latex.artofproblemsolving.com/e/b/3/eb384916db4e019c7ba11f52fadf973f515891e2.png)
where 
denotes the Euclidean norm.
defined on 
by![\[f(x, y) = x^4 + y^4 - 2(x - y)^2.\]](http://latex.artofproblemsolving.com/1/d/9/1d9e85722971f67e3ba2dee0e12d119ceabc671a.png)
Show that there exist
(and determine them) such that![\[\forall (x, y) \in \mathbb{R}^2, f(x, y) \geq \alpha \| (x, y) \|^2 + \beta,\]](http://latex.artofproblemsolving.com/e/b/3/eb384916db4e019c7ba11f52fadf973f515891e2.png)
where 
denotes the Euclidean norm.
on the circle 
and write 
.
.
hence 
.
for all 
hence 
.
we have 
and with 
we find 
.
.
.
