The title must be: a + b = 2

(1st condition): a \(\le\) 0, b \(\ge\) 0

=> \(a^4\ge0;b^4\ge0\)

We have: a + b = 2 with \(a^4;b^4\in Z\)+

=> \(a^4+b^4\ge a+b\ge2\) (proved)

(2nd condition): a \(\ge\) 0; b \(\le\) 2: Do exactly as the 1st condition (proved)

So ............

The title must be: a + b = 2

(1st condition): a ≤

 0, b ≥

 0

=> a4≥0;b4≥0

We have: a + b = 2 with a4;b4∈Z

+

=> a4+b4≥a+b≥2

 (proved)

(2nd condition): a ≥

 0; b ≤

 2: Do exactly as the 1st condition (proved)

So ............