A math problem concerned about monotonicity
Given that is a monotonously increasing function and . Now if for certain real numbers , , , compare and .
Here we will prove that $latex x_1+x_2>1$.
Obviously, when , while the equalities don’t hold simultaneously, we can easily deduce the condition; at the same time we can get .
While if and , apparently we have
, , so , so .
Now consider if and . Thus we have and . So at the same time we have
Now if , then , then .
So we have
Which indicates that the condition holds only if .
Symmetrically we can prove the same when and .