I have to determine for all $a,b \in \mathbb{R}$, for which the function $f: \mathbb{R} \rightarrow \mathbb{R}$,$$f(x) = x^3+ax^2 +bx$$ is a diffeomorphism.
This is how far I got: I know that det $J_f$(Jacobi-matrix) can't be zero. So I tried to find the Jacobi Matrix: $$J_{ij} = \frac{\partial f_i}{\partial x_i} = 3x^2+ax+b$$ Is that correct? Since it is one dimensional I'm not sure how to calculate the Jacobi-matrix. And how do I find out the determinant? Is it just $3x^2+2ax+b$? And the function is a diffeomorphism as long as the determinant is not zero?
Thank you for your help!
We have $f( \mathbb R)= \mathbb R$. Why ?
Hence $f$ is surjective.
Now you have to determine $a$ and $b$ such that the derivative $f'$ has no zero in $\mathbb R$.
We have $f'(x)=3x^2+2ax+b$. Can you proceed ?