Something Different

Consider a family of hyperbolas given byxy = cfor different values of c. We will look only at the branches of these hyperbolas in the first quadrant --xandyare both positive.

These curves are symmetric with respect toxandy; each curve is part of a rectangular hyperbola with asymptotes x = 0 and y = 0.

Now, consider the linex + y = 2mgraphed on these same axes. The xy = c curve with the greatest value ofchaving a single point in common with the line is where we havex = y = m, that is, the point (m, m).