Yes, if the problem were just to find numbers x, y, and z, satisfying [itex]x^2+ y^2+ z^2= r^2[/itex], that maximize 8xyz, then negative values would also be acceptable. There would, in fact, be 8 different solutions.theBEAST said:Homework Statement
Here is the problem, the solution and my question (in red):
I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject anything right?
Thanks!
An optimization program using Lagrange multipliers is a mathematical technique used to solve constrained optimization problems. It involves finding the maximum or minimum value of a function subject to constraints, using a set of equations called the Lagrangian equations. This method is commonly used in economics, engineering, and other fields.
The Lagrange multiplier method involves finding the partial derivatives of the objective function and the constraints, and then setting them equal to each other. This creates a system of equations that can be solved to find the optimal values for the variables. The Lagrange multiplier, which is a scalar value, is used to account for the constraints and ensure that the solution is valid.
One advantage of using Lagrange multipliers is that it can handle both equality and inequality constraints. This means that it is a versatile method that can be applied to a wide range of optimization problems. Additionally, it provides a way to incorporate constraints into the optimization process without having to modify the objective function.
One limitation of the Lagrange multiplier method is that it can only be used for differentiable functions. This means that it may not be applicable to all types of optimization problems. Additionally, it may not always give the global optimal solution, but rather a local optimum.
Yes, the Lagrange multiplier method can be extended to handle multiple constraints. This involves creating a Lagrangian function that includes all of the constraints and using the partial derivatives of this function to find the optimal solution. This method is also known as the Lagrangian multiplier method or the method of undetermined multipliers.