Question

For functions \(f(x, y)=x+4 y\) and \(g(x, y)=x^{2}+y^{2}-1\) write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f\) subject to the constraint \(g(x, y)=0\)

Short answer

Question: Determine the conditions that must be satisfied by a point to maximize or minimize the function \(f(x, y)=x+4y\) subject to the constraint \(g(x, y)=x^2+y^2-1=0\) using the Lagrange multiplier method. Answer: The conditions that must be satisfied by a point to maximize or minimize \(f(x, y)=x+4y\) subject to the constraint \(g(x, y)=x^2+y^2-1=0\) using the Lagrange multiplier method are: \(1 = \lambda (2x)\) \(4 = \lambda (2y)\) \(x^2 + y^2 - 1 = 0\)

Step-by-step solution

Compute the Gradient of f(x, y)

Compute the gradient of the function \(f(x, y)\). In this case, we have: \(\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (1, 4)\)

Compute the Gradient of g(x, y)

Compute the gradient of the constraint function \(g(x, y)\). In this case, we have: \(\nabla g(x, y) = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) = (2x, 2y)\)

Write the Lagrange Multiplier Condition

Write the Lagrange multiplier condition, which involves equating the gradient of \(f(x, y)\) and the gradient of \(g(x, y)\). This will give us: \(\nabla f(x, y) = \lambda \nabla g(x, y)\)

Write the Lagrange Multiplier Equations

Now we can write out the Lagrange multiplier equations explicitly. In this case, we have: \(1 = \lambda (2x)\) \(4 = \lambda (2y)\)

Express the Constraint Equation

We express the constraint equation \(g(x, y)=0\). In this case, we have: \(x^2 + y^2 - 1 = 0\) So, the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f(x, y)\) subject to the constraint \(g(x, y)=0\), are: \(1 = \lambda (2x)\) \(4 = \lambda (2y)\) \(x^2 + y^2 - 1 = 0\)

Key concepts

Gradient

When working with functions, the gradient is another powerful tool that helps us understand how a function changes by providing a direction. It's essentially a vector that shows the direction of the steepest increase of the function.

In the context of Lagrange multipliers, the gradient is crucial because:

• It is used to find points where a function has extrema (maximum or minimum values) under constraints.
• It allows us to visualize changes in the function as we move in the plane or space.

For a two-variable function like \(f(x, y)\), the gradient \(abla f(x, y)\) is found by taking two partial derivatives:

• \(\frac{\partial f}{\partial x}\): partial derivative with respect to \(x\)
• \(\frac{\partial f}{\partial y}\): partial derivative with respect to \(y\)

In our solution, for \(f(x, y) = x + 4y\), the gradient is \(abla f(x, y) = (1, 4)\), indicating movement is steepest when changing \(x\) by 1 unit and \(y\) by 4 units. This gradient sets the stage for the Lagrange multiplier method, searching for optimal points.

Constraint Equation

A constraint equation is a condition that must be satisfied while searching for the maximum or minimum of a function. In the Lagrange multiplier method, it's what limits the space within which we search for extrema given by another function called the constraint function.

Constraints come in the form \(g(x,y) = 0\), representing boundaries like circles or lines within which we're searching for optimal points of \(f(x, y)\).

For the given exercise, the constraint equation is \(g(x, y) = x^2 + y^2 - 1\). This equation represents a circle with radius 1 centered at the origin. Key points regarding constraint equations:

• Defines the region or path where solutions must lie.
• Involves independent variables, \(x\) and \(y\), transformed in a specific condition.
• Works with gradients to find a balance point for our function.

Utilizing the constraint equation, we ensure the solution isn't just any point, but one that adheres to our specified condition, focusing the search on relevant possibilities.

Maximization and Minimization

Maximization and minimization are mathematical processes used to find the greatest or least values a function can take, especially under some constraints. This is pivotal in many fields, such as economics, engineering, and physics, where optimal solutions are vital.

The Lagrange multiplier technique is a popular method used to identify these extrema when constraints are present. Let's break down how this works:

• The goal is to find where the gradients of our function and constraint align, indicating potential maxima or minima.
• For \(f(x, y)=x+4y\), the function is tested under the constraint \(g(x, y)=x^{2}+y^{2}-1\).
• We use the Lagrange multiplier condition \(abla f = \lambda abla g\) alongside the constraint \(g(x,y)=0\) to derive potential points of interest.
• This condition results in equations to solve for \(\lambda\) and the variables \(x\) and \(y\).

By solving these equations:

• We identify candidate points for maximum and minimum values.
• These candidates are verified against the constraint equation to ensure they are valid within the defined region.

Thus, maximization and minimization in constrained settings reveal the best values while accounting for necessary limitations, yielding solutions grounded in real-world conditions.