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With the follow definition

Definition: We say that the function $f(x_1, \cdots ,x_n)$ is positive-homogeneous of degree $k$ in $x_1, \cdots, x_n$ if
$$ f(\lambda x_1, \cdots, \lambda x_n) = \lambda^kf(x_1, \cdots ,x_n) $$ for every $\lambda > 0$

Prove the Euler's theorem

If $f(x_1, \cdots ,x_n)$ is continuously differentiable and positive-homogeneous of degree k, then
$$ \sum_{i=1}^n \frac{\partial f}{\partial x_i} x_i = kf $$

I don't need a complete proof, I only need a any hint.

I appreciate any help


marked as duplicate by Henning Makholm, user10354138, Dietrich Burde, Lee David Chung Lin, YuiTo Cheng Jun 27 at 0:17

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  • $\begingroup$ Even thought the duplicate target is not a good post in itself, it indeed provides a hint. $\endgroup$ – Lee David Chung Lin Jun 27 at 0:13

Hint: Differentiate with respect to $\lambda$ in $$ f(\lambda x_1,\lambda x_2,\dots,\lambda x_n)=\lambda^k f(x_1,x_2,\dots,x_n). $$

  • 2
    $\begingroup$ ...and then choose and substitute a special value of $\;\lambda\;$.... $\endgroup$ – DonAntonio Jun 26 at 17:51

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