EDB — 1GS

Exercises

1. [1GS] Note:Written exam, July 4th 2018.The figure 6 shows the set $E=\{(x,y):y{e}^x+x{e}^y=1\}$.

   Properly prove the following properties:

   1. at every point $(x_0,y_0)\in E$ the assumptions of the implicit function theorem are satisfied;

   2. $E\cap \{(x,y):x>0\}$ coincides with the graph, in the form $y=f(x)$, of a single function $f$ defined on $(0,+\infty )$;

   3. $E$ is connected;

   4. $\underset{x\to +\infty }{lim}f(x)=0$.

   5. Show (at least intuitively) that $x_0>0$ exists with the property that $f$ is decreasing for $0<x<x_0$, increasing for $x>x_0$.

   

   Solution 1

   [1GV]↺↻