Question

find maximum and minimum values of the function f=2(x^2-y^2)-x^4+y^4​

Answer 1

answer.

2*(x^2-y^2) - (x^2+y^2)(x^2 - y^2)

=(x^2-y^2)(2- x^2 -y^2)

For any reasonable y, if x-> infinity, (x^2-y^2), we find that (x^2-y^2) tends to infinity and (2+x^2+y^2) tend to - infinity.

Hence at x->inf, we have the whole function resulting to - infinity. Imagine y to have any real value.

Similarly take y -> -inf , with any reasonable x. Then (x^2-y^2)-> -inf, and also ( 2 + x^2 +y^2) -> - inf. Hence the highest value is achieved here.