Solution Found!
xy versus yx Consider positive real numbers x and y.
Chapter 8, Problem 90AE(choose chapter or problem)
\(x^y\) versus \(y^x\) Consider positive real numbers x and y. Notice that \(4^3\ <\ 3^4\), while \(3^2\ >\ 2^3\), and \(4^2\ =\ 2^4\). Describe the regions in the first quadrant of the xy-plane in which \(x^y\ >\ y^x\) and \(x^y\ <\ y^x\). (Hint: Find a parametric description of the curve that separates the two regions.)
Questions & Answers
QUESTION:
\(x^y\) versus \(y^x\) Consider positive real numbers x and y. Notice that \(4^3\ <\ 3^4\), while \(3^2\ >\ 2^3\), and \(4^2\ =\ 2^4\). Describe the regions in the first quadrant of the xy-plane in which \(x^y\ >\ y^x\) and \(x^y\ <\ y^x\). (Hint: Find a parametric description of the curve that separates the two regions.)
ANSWER: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
