Inequalities with Strange Equality Conditions
This page serves to debunk the myth that
An expression attains its maximum or minimum when all (some) of the variables are equal.
If \(a, b\) are real numbers such that \( a + b = 4 \), what is the minimum value of \[\]
\[\big[(a-1) ( b-1)\big]^2?\]
Clearly, since the expression is a perfect square, the minimum that it can attain is 0. This can indeed be attained, say with \( a = 1, b=3 \). The minimum does not occur when \( a = b = \frac{4}{2} \), even though the expression is symmetric. \(_\square\)
Some inequalities are less obvious and they do not have equality case when all variables are equal.
If \(x,y,z\ge 0\) satisfy \(x+y+z=3\), find the maximum value of \(x^2y+y^2z+z^2x\).
Setting \(x=y=z=1\) gives \(x^2y+y^2z+z^2x=3\). However, is this the maximum value?
In fact, setting \(x=0, y=2, z=1\) gives \(x^2y+y^2z+z^2x=4,\) which is the actual maximum value. \(_\square\)
Try your hand at the list of problems below. Be warned that these problems are hard to (properly) solve, and could require a lot of ingenuity.
