Global max of a multivar func when restriced to a given plane.

In summary, the conversation discusses a problem on the Virginia Tech test that involves proving an inequality involving three variables. The approach involves considering a function and restricting it to a specific plane, and then using the method of Lagrange multipliers to find the optimal points on that plane. This method is an extension of concepts learned in calculus 3.
  • #1
TylerH
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On this year's Virginia Tech test there was a problem to prove:
[itex]\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}\leq \frac{3\sqrt{3}}{2}[/itex]
when
[itex]x+y+z=xyz[/itex]

So my approach was to consider the function, f, where
[itex]f(x, y, z)=\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}[/itex]
then restrict f to the plane x+y+z=xyz, then maximize f.

How could I restrict f to that plane and then maximize?
 
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  • #2
The solution set of x + y + z = xyz is not a 2-dimensional plane. It is a level set of the function g(x, y, z) = x + y + z - xyz. Let's call that set of points M. To optimize a function subject to a constraint of this type, it is best to use the method of Lagrange multipliers. In particular, if P = (a, b, c) is a stationary point for f over the manifold M, then the gradient of f at P must be a scalar multiple λ of the gradient of g at P. This equation, plus the fact that g(P) = 0 provides a system of 4 equations in 4 unknowns that you can solve for the requisite points. You can then use the Hessian to determine the identity of these points.
 
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  • #3
slider142 said:
The solution set of x + y + z = xyz is not a 2-dimensional plane. It is a level set of the function g(x, y, z) = x + y + z - xyz. Let's call that set of points M. To optimize a function subject to a constraint of this type, it is best to use the method of Lagrange multipliers. In particular, if P = (a, b, c) is a stationary point for f over the manifold M, then the gradient of f at P must be a scalar multiple λ of the gradient of g at P. This equation, plus the fact that g(P) = 0 provides a system of 4 equations in 4 unknowns that you can solve for the requisite points. You can then use the Hessian to determine the identity of these points.
Oh! That's actually a sort of natural extension to the 3d stuff I learned in calc 3. Thanks!
 

1. What does it mean to find the global maximum of a multivariable function when restricted to a given plane?

Finding the global maximum of a multivariable function when restricted to a given plane means finding the highest possible value of the function within a specific 2-dimensional space. This can be visualized as finding the highest point on a curved surface when looking at it from a specific angle or perspective.

2. How is the global maximum of a multivariable function different from the local maximum?

The global maximum of a multivariable function is the highest value of the function within a given domain, while the local maximum is the highest value of the function within a specific region. The global maximum may be located at a point where the function is not differentiable, while the local maximum is always located at a point where the function has a critical point.

3. What is a constraint in a multivariable function and how does it affect the global maximum?

A constraint in a multivariable function is a limiting factor or condition that must be satisfied in order to find the optimal solution. In the context of finding the global maximum, a constraint may be a restriction to a specific plane or a limitation on the values of the variables. Constraints can affect the global maximum by reducing the possible solutions and making it more difficult to find the optimal value.

4. How do we find the global maximum of a multivariable function when restricted to a given plane?

To find the global maximum of a multivariable function when restricted to a given plane, we can use techniques such as Lagrange multipliers or substitution to eliminate one variable and solve for the other(s). We can also use graphical methods, such as contour plots, to visualize the function and identify the maximum point. In some cases, we may need to use numerical methods, such as gradient descent, to approximate the global maximum.

5. Can a multivariable function have multiple global maxima when restricted to a given plane?

Yes, it is possible for a multivariable function to have multiple global maxima when restricted to a given plane. This can occur when the function has multiple critical points that all satisfy the constraint. In some cases, there may be an infinite number of global maxima along the boundary of the constraint. It is important to carefully consider the constraints and the behavior of the function in order to accurately identify all possible global maxima.

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