Solving f(x,y,z) with x,y,z Constraints

  • Thread starter DrunkApple
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In summary, the conversation is about finding the correct boundaries for integrating the function f(x,y,z) = e^(2x-z) over the region defined by x+y+z ≤ 1 and x,y,z ≥ 0. The integrand can be factored into e^(2x) e^(-z) and the boundaries are symmetric under permutations of x, y, and z. The person is struggling to set up the boundaries correctly and asks for help from others to show their attempts. They also mention that they can provide their work if needed.
  • #1
DrunkApple
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Homework Statement


f(x,y,z) = e^(2x-z)
W: x+y+z ≤ 1, x,y,z ≥ 0

Homework Equations


0 ≤ x ≤ 1-y-z
0 ≤ y ≤ 1-x-z
0 ≤ z ≤ 1-x-y

The Attempt at a Solution


I tried dzdydx and dydzdx but they don't work...
or am I doing something wrong?
 
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  • #2
You will note that the boundaries are symmetric under permutations of x, y, and z. Thus w.r.t. the boundary order of integration shouldn't matter.

The integrand will be quite easy to integrate. I suggest you factor it into e^(2x) e^(-z).

You just have to set up your boundaries of integration correctly. Can you show us some of your attempts?
 
  • #3
Isn't that the correct boundaries that I set up in relevant equations?
or is that wrong?
I would love to post my work... but it's so much mess right now... it's like everywhere... If I can get the right boundaries I feel like I can get it right
 
  • #4
Give me what you got...

integral from what to what, from what to what, from what to what of what dwhatdwhatdwhat?
 

1. What is the purpose of solving f(x,y,z) with x,y,z constraints?

The purpose of solving f(x,y,z) with x,y,z constraints is to find the values of x, y, and z that will satisfy the given function f. Constraints are used to limit the possible solutions and make the problem more manageable.

2. How do constraints affect the solution for f(x,y,z)?

Constraints can greatly impact the solution for f(x,y,z) by limiting the possible values of x, y, and z. This can make the problem easier to solve and can also provide a more accurate and meaningful solution.

3. What are some common types of constraints used in solving f(x,y,z)?

Some common types of constraints used in solving f(x,y,z) are inequalities, equations, and boundary conditions. These constraints can be applied to one or more of the variables x, y, and z to limit their possible values.

4. How does changing the constraints affect the solution for f(x,y,z)?

Changing the constraints can have a significant impact on the solution for f(x,y,z). By altering the constraints, the range of possible solutions may change, and the optimal solution may also change. In some cases, changing the constraints may even result in no feasible solution.

5. Can constraints be added or removed during the solving process for f(x,y,z)?

Yes, constraints can be added or removed during the solving process for f(x,y,z). This can be useful in cases where the initial set of constraints does not provide a feasible solution. By adding or removing constraints, the problem can be adjusted to find a suitable solution.

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