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

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The discussion focuses on the function f(x,r) = √x - rx, where x is non-negative, and the challenge of understanding how the value function f*(r), representing the maximum of f(x,r) for each r, acts as an envelope for the various (x,r) functions. It is clarified that while f(x,r) is linear in r for fixed x, it is nonlinear in x, necessitating the identification of the optimal x* that maximizes f(x,r) for each r. The key takeaway is that f*(r) is constructed by evaluating f at this optimal x*, thus serving as an envelope for the family of functions defined by varying x. This relationship illustrates the envelope theorem's application in nonlinear contexts. Understanding this maximization process is crucial for correctly sketching both f(x,r) and f*(r).
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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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