The problem consists of two questions :
The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that $$t_x^3x^2+t_x+x=0$$
I'm having problems with the second question which asks to study the implicit function $f:x\longmapsto t_x$
A hint asked to study the inverse function of $f$ but I'm having a hard time to prove that $f$ is invertible, let alone to study its inverse.
What I could do :
- $f(x)$ and $x$ have opposite signs due to the equation $t_x(x^2t_x^2+1)=-x$
- seeing the equation as a quadratic in $x$, the discriminant should be $\ge 0$ meaning range of $f$ is within $\left[-\frac{1}{\sqrt 2}, \frac{1}{\sqrt 2} \right]$
- if $f$ is invertible the inverse has one of the expressions $\dfrac{-1\pm\sqrt{1-4t^4}}{2t^3}$ (got by solving $(g(y))^2y^3+g(y)+y=0$)
- A few tests on wolframalpha show that $f$ would be odd, negative and decreasing on $[0,+\infty)$.
Any help or recommendations to treat such problems would be appreciated.