A function $f : R^n → R$ is called homogeneous of degree D if for every $λ ∈ R$ and $(x_1, . . . , x_n) ∈ R^n $ we have the equality $f(λx_1, λx_2, . . . , λx_n) = λ Df(x_1, x_2, . . . , x_n)$. Show that any homogeneous differentiable function $f$ of degree $D$ satisfies $\sum_{k=1}^{n}x_k\frac{\partial f}{\partial x_k} > = D · f ∀ (x_1, x_2, . . . , x_3) ∈ R^n.$
How do I do this question. I do not know where to start
I am not quite sure exactly what you are asking but the following may help you on your way.
Since $f$ is homogeneous of degree $d$, you have $f(\lambda x_1,\dots,\lambda x_n)=\lambda^d f(x_1,\dots,x_n)$. Defining $g(\lambda,x_1,\dots,x_n):=f(\lambda x_1,\dots,\lambda x_n)$ you can compute \begin{equation} \frac{\partial g}{\partial\lambda} = \sum_{i=1}^n x_i\frac{\partial f}{\partial x_i}, \end{equation} or instead writing $g(\lambda,x_1,\dots,x_n)=\lambda^d f(x_1,\dots,x_n)$ you can compute \begin{equation} \frac{\partial g}{\partial \lambda} = d\lambda^{d-1}f(x_1,\dots,x_n). \end{equation}
These two expressions for $\partial g/\partial\lambda$ may be useful.