Problem 16 - Calculus an Applied Approach

Question

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y,\) and \(z\) are positive. $$ \begin{array}{l}{\text { Minimize } f(x, y)=x^{2}-8 x+y^{2}-12 y+48} \\\ {\text { Constraint: } x+y=8}\end{array} $$

Step-by-Step Solution

Answer

The minimum value of the function \(f(x, y)\) subject to the constraint \(x+y=8\) is -17.

Step 1 - Setup the Lagrange system

We introduce a new variable, \(\lambda\), known as a Lagrange multiplier. We set \[ L(x, y, \lambda) = f(x,y) - \lambda(g(x,y)-c) = x^{2}-8 x+y^{2}-12 y+48 -\lambda (x+y-8) \] where \(L\) is the Lagrangian, \(f\) is the function to optimize, \(g(x,y)=x+y\) is the constraint function, and \(c=8\) is the required sum of \(x\) and \(y\).

Step 2 - Differentiating the equations

Differentiate \(L\) with respect to \(x\), \(y\) and \(\lambda\), setting each derivative equal to zero. This gives us the following system of equations: 1. \[ \frac{\partial L}{\partial x} = 2x-8-\lambda =0 \]2. \[ \frac{\partial L}{\partial y} = 2y-12-\lambda = 0 \]3. \[ \frac{\partial L}{\partial \lambda} = x+y-8 = 0 \]

Step 3 - Solving the system of equations

After defining these three equations, the next step is to solve them. 1. The third equation, \(x+y=8\), is already the constraint equation. 2. From the first equation, we can express \(\lambda\) in terms of \(x\) ie \(\lambda = 2x-8\).3. Substituting this into the second equation we get \(2y-12=2x-8 \rightarrow y=x+2\). Substitute \(y=x+2\) back into the constraint equation to solve for \(x\), we get \(x+ (x+2)=8 \rightarrow x=3\) and \(y=5\).

Step 4 - Finding the minimum value

Substitute these values of \(x\) and \(y\) into the function \(f\), we get: \[f(3,5)=3^{2}-8*3+5^{2}-12*5+48 = -17.\]

Key Concepts

constrained optimization

Constrained optimization involves finding the extremum (maximum or minimum) of a function, given some restrictions or constraints. In our exercise, the constraint is that the sum of the variables, \(x\) and \(y\), must equal 8.

To solve this, we use Lagrange multipliers, a technique that introduces an auxiliary variable, \(\lambda\), to account for the constraint. This helps transform the problem into finding the extremum of a new function, the Lagrangian, which incorporates both the original function and the constraint.

In real-world applications, constraints can be physical limits, cost limitations, or other restrictions. Lagrange multipliers are especially valuable in economics and engineering, where such boundary conditions often occur.

calculus optimization

Calculus optimization encompasses techniques used to find the extremum of functions. It leverages differential calculus principles to identify points where a function reaches its highest or lowest value. This involves taking derivatives and setting them to zero to find critical points.

In the context of our problem, the function to minimize is \(f(x, y) = x^{2} - 8x + y^{2} - 12y + 48\). Without constraints, we'd differentiate independently with respect to \(x\) and \(y\) and solve for zero slopes. However, due to the constraint \(x + y = 8\), we use the Lagrangian approach, differentiating simultaneously with respect to \(x\), \(y\), and \(\lambda\).

This adds an extra layer to basic calculus optimization, allowing solutions for more complex systems where variables cannot change independently.

extremum finding

Finding the extremum of a function, such as a minimum or maximum, is a key objective in optimization problems. In our exercise, the goal is to minimize \(f(x, y)\) given a constraint. The solution reveals that the minimum occurs at \(x = 3, y = 5\).

To ensure an extremum is found, we set the partial derivatives of the Lagrangian to zero and solve for the variables. This method relies on the necessary conditions that the derivatives are equal to zero at the extremum point.

In broader mathematical and practical applications, extremum finding helps in areas like resource allocation, production optimization, and even computer algorithms, wherever efficiency and cost-saving are prioritized.

mathematical problem solving

Mathematical problem solving involves employing different mathematical techniques and tools to find solutions to given problems. Using Lagrange multipliers to handle constrained problems is a quintessential approach.

The exercise illustrates key problem-solving steps:

• Define the goal clearly, here minimizing \(f(x, y)\).
• Incorporate constraints, resulting in the formation of the Lagrangian.
• Differentiation to reveal critical points, leading to the system of equations.
• Solve systematically to find the solution that satisfies all requirements.

This structured approach allows for tackling complex problems effectively, aiding not just in academic settings but in real-world problem-solving too, where methodical analysis is crucial.