Hello, all. Digging though my last home assignment here, and stuck on 2 problems once again. Here's the first one:
Prove that a^2 + b^2 + 1 > a + b + ab.
I'm considering two approaches. First one is to consider all cases:
When a and b are < 0, the statement is obviously true, since left side will always be >0 and right side will always be < 0 When a and b are equal and are equal to 0, the left side is greater, since it has a constant. When a and b are equal and are equal to 1, both sides are the same, and thus the equation still holds. My only troubles are when a and b are between 0 and 1, and when they are greater than one. Cannot find a proof without algebraic manipulation.
Second approach: algebraically prove the ineqality. I rewrote the inequality as a^2 + b^2 + 2ab > a + b + 3ab -1, leading to (a + b)^2 > a + b + 3ab -1. This has not lead me to any meaningful solution. I also tried writing as a^2 - a + b^2 - b > ab -1, leading to a(a-1) + b(b-1) > ab - 1. Stuck again.