Math Problem Solver

Prove Euler's Theorem: Suppose that $g: \mathbb{R}^{K+2} \rightarrow \mathbb{R}$ is differentiable in $x \in \mathbb{R}$ and $y \in \mathbb{R}$ with partial derivatives denoted by $g_{x}$ and $g_{y}$, and is homogeneous of degree $m$ in $x$ and $y$. Then

$$m g(x, y, z)=g_{x}(x, y, z) x+g_{y}(x, y, z) y \quad \forall x \in \mathbb{R}, y \in \mathbb{R}, \text { and } z \in \mathbb{R}^{K}$$

Moreover, $g_{x}(x, y, z)$, and $g_{y}(x, y, z)$ are themselves homogeneous of degree $m-1$ in $x$ and $y$.

Solve with each and every step, do not skip any step