How Does the Envelope Theorem Apply to Nonlinear Functions Like f(x,r)?

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SUMMARY

The discussion centers on the application of the Envelope Theorem to the function f(x,r) = √x - rx, where x ≥ 0. Participants clarify that while f(x,r) is linear in terms of r, it is nonlinear with respect to x. The key takeaway is that to construct the envelope function f*(r), one must identify the value x* that maximizes f(x,r) for each fixed r, thereby demonstrating how f*(r) serves as an envelope for the family of functions defined by varying x.

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Hi. I'm trying to solve the following problem; am confused.

Define the function f(x,r) = √x - rx, where x ≥ 0. Sketch both the function for several values of x, and the value function f*(r), which is the maximum value of f(x,r) for each given value of r. Describe how the function f*(r) is an envelope of the different (x,r) functions.

But f(x,r) is linear for given x - how can f*(r) envelope linear functions? Can anyone help?
 
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Peterw222 said:
Hi. I'm trying to solve the following problem; am confused.

Define the function f(x,r) = √x - rx, where x ≥ 0. Sketch both the function for several values of x, and the value function f*(r), which is the maximum value of f(x,r) for each given value of r. Describe how the function f*(r) is an envelope of the different (x,r) functions.

But f(x,r) is linear for given x - how can f*(r) envelope linear functions? Can anyone help?

Welcome on MHB Peterw222!...

... the function...

$\displaystyle f(x,r) = \sqrt{x} - r\ x\ (1)$

... is linear in r and non linear in x... what You have to do is, given r, find the value x* that maximizes (1) and construct f*(r) = f(x*,r)...

Kind regards

$\chi$ $\sigma$
 

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