3.II.13B

Analysis II | Part IB, 2005

Let ff be a real-valued differentiable function on an open subset UU of Rn\mathbf{R}^{n}. Assume that 0U0 \notin U and that for all xUx \in U and λ>0,λx\lambda>0, \lambda x is also in UU. Suppose that ff is homogeneous of degree cRc \in \mathbf{R}, in the sense that f(λx)=λcf(x)f(\lambda x)=\lambda^{c} f(x) for all xUx \in U and λ>0\lambda>0. By means of the Chain Rule or otherwise, show that

Dfx(x)=cf(x)\left.D f\right|_{x}(x)=c f(x)

for all xUx \in U. (Here Dfx\left.D f\right|_{x} denotes the derivative of ff at xx, viewed as a linear map RnR\mathbf{R}^{n} \rightarrow \mathbf{R}.)

Conversely, show that any differentiable function ff on UU with Dfx(x)=cf(x)\left.D f\right|_{x}(x)=c f(x) for all xUx \in U must be homogeneous of degree cc.

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