EDB — 1RV

[1RV]Let’s start with the first definition. Suppose we can apply the Implicit Function Theorem to the locus

Precisely, suppose that at a point \((\overline x,\overline y,\overline a)\) we have that \(\frac{\partial {}F}{\partial {y}} ≠ 0\). To this we also add the hypothesis \(\frac{\partial {}^ 2F}{\partial {aa}}≠ 0\). Fixed \(a\), you can express \(E_ a\) locally as a graph \(y=f(x,a)=f_ a(x)\). We also use the hypothesis \(\frac{\partial {}^ 2F}{\partial {aa}}≠ 0\) to express locally \(\frac{\partial {}F}{\partial {a}}= 0\) as a graph \(a=Φ(x,y)\). Defining \(G(x,y){\stackrel{.}{=}}F(x,y,Φ(x,y))\), show that \(G=0\) can be represented as \(y=g(x)\). Finally, show that \(g\) is the envelope of the curves \(f_ a\).