EDB — 1GS

Exercises

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

   Properly prove the following properties:

   1. at every point \((x_ 0,y_ 0)∈ E\) the assumptions of the implicit function theorem are satisfied;

   2. \(E∩\big\{ (x,y): x{\gt}0\big\} \) coincides with the graph, in the form \(y=f(x)\), of a single function \(f\) defined on \((0,+∞)\);

   3. \(E\) is connected;

   4. \(\lim _{x→+∞} f(x)=0\).

   5. Show (at least intuitively) that \(x_ 0{\gt}0\) exists with the property that \(f\) is decreasing for \(0{\lt}x{\lt}x_ 0\), increasing for \(x{\gt}x_ 0\).

   

   Solution 1

   [1GV]↺↻