Let $f: \mathbb{R^2} \to \mathbb{R}$ with $f(x,y) := 6y+4x-x^3y$
How can one argue that $f$ is infinitely often continuously partially differentiable?
I also calculated the gradient $\nabla f: \mathbb{R^2} \to \mathbb{R^2}$ and got
$\nabla f = (4-3yx^2, 6-x^3)$
The Hessian matrix $H_f: \mathbb{R^2} \to \mathbb{R}^{2 × 2}$ is
$H_f = \begin{pmatrix} -6yx& -3x^2 \\ -3x^2& 0 \end{pmatrix}$
Can someone confirm that and the question above? Thanks!