Problem 90AE

xy versus yx Consider positive real numbers x and y. Notice that 43<34, while 32 > 23, and 42 = 24. Describe the regions in the first quadrant of the xy-plane in which xy > yx and xy<yx. (Hint:Find a parametric description of the curve that separates the two regions.)

Solution 90AE

Consider the region shown in the figure (1):

Consider the region shown in the figure (2):

Notice that while and

So by observing the conditions while conclude that

And the curve separates the two regions and

And clearly we know that is true when

The parametric representation for the line is

Also the parametric representation for the remaining part of the graph of the equation is,

Also there is a hole at the point, so doesn’t include the point in the parametrization.

The complete curve can be represented by

Step 4 of 4

The graph of the curve and the two regions and are shown in the figure 3: