Application of implicit theorem

Question

Find conditions on the function $f$ and $g$ which permit you to solve the equations $$f(xy)+g(yz)=0\ \ \textrm{and} \ \ g(xy)+f(yz)=0 $$ for $y$ and $z$ as functions of $x$, near the point $x=y=z=1$ and $f(1)=g(1)=0$.

Attempt: This problem seems to be an application of the implicit function theorem. Usually on this kind of problems we define special transformations, but I don't now how to define it in this case since I have several variables $x,y,z$ and two functions $f$ and $g$. Any idea on how to start?

Perhaps try looking at $f(s)+g(t) = 0, f(t)+g(s)= 0$ near $s=t=1$ first? — copper.hat, Feb 13 '21 at 05:08

score 1 · Accepted Answer · answered Feb 13 '21 at 05:16

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Let $F((x,y,z)) = \begin{bmatrix} f(xy) + g(yz) \\ f(yz)+g(xy)\end{bmatrix}$ and show that $Df((1,1,1)) = \begin{bmatrix} f'(1) & f'(1)+g'(1) & g'(1) \\ g'(1) & f'(1)+g'(1) & f'(1)\end{bmatrix}$. Pick any $2 \times 2$ submatrix and find a condition that ensures that the submatrix is invertible.

answered Feb 13 '21 at 05:16

copper.hat

172,524

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This idea is very clear! Helps a lot – apa Feb 13 '21 at 05:39

Application of implicit theorem

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