Two Handy Inequalities
This problem was suggested by Alexander Karabegov.
Prove the following inequalities are true for all positive real numbers x and y, with equality holding if and only if x = y.
Hint: (x–y)2 is greater than or equal to 0.
Please submit all work to mathpotw@acu.edu no later than 5:00 PM on Thursday, October 17.
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Solution to Two Handy Inequalities
In fact, we can prove a bit more than the problem asked. We will show that
with either equality holding if, and only if, x = y.
It is certainly true that (x–y)2 > 0 when, and only when, x is not equal to y. From this expression we derive the inequality
x2 + y2 > 2xy (*)and from this inequality, we derive two more inequalities.
First, by adding x2 + y2 to both sides of inequality (*), we find that 2(x2+y2) > x2 + 2xy + y2.
Next, by adding 2xy to both sides of the expression (*), we see that x2 + 2xy + y2 > 4xy.
The claimed inequalities follow easily from the two expressions we just derived.
