What is the derivative of max(u(x),v(x)) when u(x) and v(x) are given functions?

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Discussion Overview

The discussion centers on determining the derivative of the function f(x) = max(u(x), v(x)), where u(x) and v(x) are given functions. Participants explore the conditions under which this function may be differentiable, considering both theoretical and practical implications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asks about the derivative of f(x) = max(u(x), v(x)).
  • Another suggests analyzing the cases when u(x) > v(x) and v(x) > u(x) as a starting point.
  • Some participants note that max(u(x), v(x)) can be differentiable even if u(x) and v(x) are not continuous, providing an example with rational and irrational values.
  • Conversely, it is also mentioned that max(u(x), v(x)) may not be differentiable even when u(x) and v(x) are continuous, citing the example of u(x) = x and v(x) = -x, leading to a non-differentiable point at x = 0.

Areas of Agreement / Disagreement

Participants express differing views on the differentiability of max(u(x), v(x)) under various conditions. No consensus is reached regarding the general behavior of the derivative in relation to the continuity of u(x) and v(x).

Contextual Notes

The discussion highlights the complexity of differentiability in piecewise functions and the influence of continuity on the derivative, but does not resolve the implications of these examples fully.

Dansuer
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What is the derivative of the function f(x)= max(u(x),v(x)) ?
where u(x) and v(x) are two given function
 
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Try looking at the two cases when u(x)>v(x) and when v(x)>u(x)
 
Office_Shredder said:
Try looking at the two cases when u(x)>v(x) and when v(x)>u(x)

That is a good place to start, but max(u(x),v(x)) can be differentiable when u(x) and v(x) are not even continuous.

For example
u(x) = 0 when x is rational, u(x) = 1 otherwise
v(x) = 1 when x is rational , v(x) = 0 otherwise
 
AlephZero said:
That is a good place to start, but max(u(x),v(x)) can be differentiable when u(x) and v(x) are not even continuous.

For example
u(x) = 0 when x is rational, u(x) = 1 otherwise
v(x) = 1 when x is rational , v(x) = 0 otherwise

Or max(u(x),v(x)) can not be differentiable, while u(x) and v(x) are:

For example:
u(x)=x and v(x)=-x

Then max(u(x),v(x))=|x| which is not differentiable in 0.
 

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