help at solving a system

Question

Given a function $f(x)=x+\sqrt[3]{x}+\sqrt[5]{x}-2$, I have to solve the following system: $$\left\{\begin{matrix} x+\sqrt[3]{x}+\sqrt[5]{x}=y+2\\ y+\sqrt[3]{y}+\sqrt[5]{y}=z+2\\ z+\sqrt[3]{z}+\sqrt[5]{z}=x+2 \end{matrix}\right.$$ Could you give me any hint? Thank you very much!

Answer 1

Hint

Using the function $f$ the system can be written as: $$\left\{\begin{matrix} f(x)=y\\ f(y)=z\\ f(z)=x \end{matrix}\right.$$ In particular: $$f(f(f(x)))=x$$ Now you can notice that: $$f(x) > x \text{ if } x>1$$ $$f(x) < x \text{ if } x<1$$ $$f(1)=1$$

Then you can prove that:

• if $x>1$ then $f(f(f(x)))>x$
• if $x<1$ then $f(f(f(x)))<x$

to conclude that the only solution is $x=y=z=1$.