Question

Maximizing a sum. Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\)

Short answer

Answer: The maximum value of x₁ + x₂ + x₃ + x₄ under the given constraint is 8, which occurs when (x₁, x₂, x₃, x₄) = (2, 2, 2, 2).

Step-by-step solution

Set up the Lagrange function

We start by setting up the Lagrange function which is \(L(x_1, x_2, x_3, x_4, \lambda) = F(x_1, x_2, x_3, x_4) - \lambda G(x_1, x_2, x_3, x_4)\). In our case, the Lagrange function will be: \(L(x_1, x_2, x_3, x_4, \lambda) = (x_1 + x_2 + x_3 + x_4) - \lambda (x_1^2 + x_2^2 + x_3^2 + x_4^2 - 16)\)

Compute the gradient of L

Now we need to find the gradient of L for critical points. Compute the partial derivatives with respect to \(x_1, x_2, x_3, x_4, \text{and } \lambda\), and set them equal to 0: \(\frac{\partial L}{\partial x_1} = 1 - 2\lambda x_1 = 0\) \(\frac{\partial L}{\partial x_2} = 1 - 2\lambda x_2 = 0\) \(\frac{\partial L}{\partial x_3} = 1 - 2\lambda x_3 = 0\) \(\frac{\partial L}{\partial x_4} = 1 - 2\lambda x_4 = 0\) \(\frac{\partial L}{\partial \lambda} = x_1^2 + x_2^2 + x_3^2 + x_4^2 - 16 = 0\)

Solve the equations found in step 2

From the partial derivatives equal to 0: \(x_1 = x_2 = x_3 = x_4 = \frac{1}{2\lambda}\) Substituting these values back into the constraint \(G(x_1, x_2, x_3, x_4) = 0\), we get: \(\frac{1}{4\lambda^2} + \frac{1}{4\lambda^2} + \frac{1}{4\lambda^2} + \frac{1}{4\lambda^2} - 16 = 0\) Solving for \(\lambda\), we find two possible values: \(\lambda = \frac{1}{4}\) or \(\lambda = -\frac{1}{4}\).

Calculate the critical points

Using the values of \(\lambda\) found in step 3, we can now calculate the critical points: For \(\lambda = \frac{1}{4}\), the critical point is \((2, 2, 2, 2)\), and the value of the objective function at this point is \(F(2, 2, 2, 2) = 2+2+2+2=8\). For \(\lambda = -\frac{1}{4}\), we get the critical point \((-2, -2, -2, -2)\). The value of the objective function is \(F(-2, -2, -2, -2) = -2-2-2-2=-8\).

Determine the maximum value

Now, we compare the values from step 4. We are looking for the maximum value of the sum, and since \(8 > -8\), the maximum value of \(x_1 + x_2 + x_3 + x_4\) under the given constraint is \(8\) which occurs when \((x_1, x_2, x_3, x_4) = (2, 2, 2, 2)\).

Key concepts

Maximization Problems

Maximization problems occur when we're aiming to find the greatest possible value of a particular function. Here, for example, we seek to maximize the sum \(x_{1}+x_{2}+x_{3}+x_{4}\). This involves looking at how these values can be adjusted, given certain limitations or constraints. It’s the process of adjusting the variables to find the most favorable outcome. In real life, maximization problems help with decisions like maximizing profits or optimizing resources.
To tackle such a problem successfully, you need to clearly identify the function you want to maximize. Then, examine any restrictions or conditions that might affect the variables involved. Often these restrictions form an equation that helps guide the maximization process. With this clear picture, you can employ techniques such as setting up an appropriate mathematical model, using derivatives, or employing specialized strategies like Lagrange multipliers to find the solution.

Constraint Optimization

Constraint optimization is a fascinating aspect of mathematical optimization in which you find the best outcome within given limitations. In this exercise, the constraint is expressed as \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\). Constraints can take various forms, such as equations or inequalities, and impose necessary conditions that solutions must adhere to.
Managing these constraints requires balancing the desire to maximize or minimize an objective with the need to satisfy other criteria. The challenge lies in working within these boundaries while finding an optimal solution. Techniques such as the Lagrange multipliers are particularly helpful here because they allow you to incorporate constraints directly into the optimization process. By adjusting both the variables of interest and the Lagrange multipliers, one can maneuver towards a solution that meets all conditions.

Gradient Calculation

Gradient calculation is a crucial step in solving maximization problems with constraints. It involves finding the gradient of a function, which gives us a vector indicating the direction of the steepest ascent. In this exercise, we calculate the gradient of the Lagrange function \(L(x_1, x_2, x_3, x_4, \lambda)\).
To find critical points where potential maxima or minima might occur, one takes the partial derivative of the Lagrange function with respect to each variable and the Lagrange multiplier \(\lambda\). Setting these derivatives equal to zero helps determine points where the function doesn’t increase or decrease, indicating possible solutions under the defined constraints. This step bridges the objective of maximizing the function and the necessity to comply with the constraint, leading to viable critical points that represent optimal solutions.