Chapter 3
Inequalities and Constrained
Extrema
3.1 A Representative Example
The thrust of this chapter can probably be summarized through the
following very simple example. Starting with the very simple observa-
tion that for real numbersxandy, 0≤(x−y)^2. Expanding the right
hand side and rearranging gives the following inequality:
2 xy ≤ x^2 +y^2 ,again valid for allx, y∈R. Furthermore, it is clear that equality ob-
tains precisely whenx=y. We often refer to the as anunconditional
inequality, to be contrasted from inequalities which are true only for
certain values of the variable(s). This is of course, analogous to the
distinction between “equations” and “identities” which students often
encounter.^1
We can recast the above problem as follows.(^1) By way of reminder, theequalityx (^2) −x−6 = 0 admits a solution, viz.,x=− 2 ,3, whereas
the equalityx(x−2) =x^2 −2 is always true (by the distributive law), and hence is anidentity.
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