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