This is a problem from a previous graduate preliminary exam in multivariable analysis/calculus that I am trying to solve for my own practice:
Problem:
Let $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ be a twice continuously differentiable function satisfying: $f\left ( 0,y \right )=0$ for all $y\in \mathbb{R}$.
1- Prove that $f\left ( x,y \right )=x.g\left ( x,y \right )$ for all $\left ( x,y \right )\in \mathbb{R}^{2}$, where: $g\left ( x,y \right )=\int_{0}^{1}\frac{\partial f\left ( tx,y \right )}{\partial x}dt$.
2- Show that $g$ is continuously differentiable, and that for all $x$ in $\mathbb{R}$:
$g\left ( 0,y \right )=\frac{\partial f}{\partial x}\left ( 0,y \right )$ and $\frac{\partial g}{\partial y}\left ( 0,y \right )=\frac{\partial ^{2}f}{\partial x\partial y}\left ( 0,y \right )$
For the first part: The only thing I could do so far is the following: I am basically trying to start from the right hand side to reach the left hand side. Note that: $g\left ( x,y \right )=\int_{0}^{1}\frac{\partial f\left ( tx,y \right )}{\partial x}dt=\int_{0}^{1}t\frac{\partial f\left ( u,y \right )}{\partial u}dt$ where $u=tx$. Then, I have no idea how to go forward?
Any help is appreciated.