Question

find tha maxima and minima of f(x,y)=x4+y4-2x2+4xy-2y2​

Answer 1

Step-by-step explanation:

It's obvious that a maximal value does not exist.

We'll prove that −8−8 is a minimal value.

Let x=2–√ax=2a and y=−2–√by=−2b.

Hence, we need to prove that

4a4+4b4−4a2−4b2−8ab+8≥04a4+4b4−4a2−4b2−8ab+8≥0

or

a4+b4+2≥(a+b)2,a4+b4+2≥(a+b)2,

which is AM-GM and C-S:

a4+b4+2≥2(a4+1)(1+b4)−−−−−−−−−−−−−√≥2(a2+b2)≥(a+b)2