Simple differentiation - Part 4

I did not know the mathematical proof behind it but just did it logically !!!! THANK you for the inequality !!!!

But isn't -290 considered to be less tan 0? hence, a=-290, b=1

@Azhaghu Roopesh M , AM-GM inequality won't work here. It gives you an upper bound for the product, not the lower bound which we want to find! A mathematical proof would be simply the multiplication of similarly oriented inequalities.

$a,b\in\mathbb{W}\implies a\geq 0~\land~b\geq 0\implies ab\geq 0~;~a+b=k~\forall~k\in\mathbb{Z^+}$

Note that multiplication of the inequalities is valid since they are similarly oriented and the equality cases occuring in the the final symmetric inequality $(\textrm{at }a=k,b=0~\textrm{or}~a=0,b=k$ ) are in accordance with the starting inequalities. Also, the problem conditions are not violated. Also, the value of the sum doesn't matter. The answer remains the same.