Problem 57 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The maximum value of the sum is \(8\).

Step 1: Set up the optimization problem with Lagrange Multipliers

First, let us set up the Lagrange function, which combines the objective function (the sum of \(x_1+x_2+x_3+x_4\)) and the constraint (\(x_1^2+x_2^2+x_3^2+x_4^2=16\)) with a Lagrange multiplier, denoted as \(\lambda\): \(L(x_1, x_2, x_3, x_4, \lambda) = x_1 + x_2 + x_3 + x_4 - \lambda(x_1^2 + x_2^2 + x_3^2 + x_4^2 - 16)\)

Step 2: Compute the partial derivatives of the Lagrange function

To find the extrema of the Lagrange function, we need to compute the partial derivatives of \(L(x_1, x_2, x_3, x_4, \lambda)\) with respect to each variable and set them to zero: \(\frac{\partial L}{\partial x_1} = 1 - 2\lambda x_1 = 0\) \(\frac{\partial L}{\partial x_2} = 1 - 2\lambda x_2 = 0\) \(\frac{\partial L}{\partial x_3} = 1 - 2\lambda x_3 = 0\) \(\frac{\partial L}{\partial x_4} = 1 - 2\lambda x_4 = 0\) \(\frac{\partial L}{\partial \lambda} = x_1^2 + x_2^2 + x_3^2 + x_4^2 - 16 = 0\)

Step 3: Solve the system of equations from the partial derivatives

Now that we have a system of 5 equations, we need to solve for \(x_1, x_2, x_3, x_4,\) and \(\lambda\). From the first four equations, we can obtain the following expressions for each variable: \(x_1 = \frac{1}{2\lambda}\) \(x_2 = \frac{1}{2\lambda}\) \(x_3 = \frac{1}{2\lambda}\) \(x_4 = \frac{1}{2\lambda}\) Substitute these expressions into the constraint equation (i.e., \(\frac{\partial L}{\partial \lambda} = 0\)): \((\frac{1}{2\lambda})^2 + (\frac{1}{2\lambda})^2 + (\frac{1}{2\lambda})^2 + (\frac{1}{2\lambda})^2 = 16\) Solve for \(\lambda\): \(\frac{4}{4\lambda^2} = 16 \Rightarrow \lambda^2 = \frac{1}{64} \Rightarrow \lambda = \frac{1}{8}\) Now, plug the value of \(\lambda\) back into the expressions for \(x_1, x_2, x_3,\) and \(x_4\): \(x_1 = x_2 = x_3 = x_4 = 2\)

Step 4: Compute the maximum value of the sum

With the values of \(x_1, x_2, x_3,\) and \(x_4\) determined, we can compute the maximum value of the sum: \(x_{1}+x_{2}+x_{3}+x_{4} = 2 + 2 + 2 + 2 = 8\) Thus, the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the given constraint is \(\boxed{8}\).

Key Concepts

Optimization

Optimization refers to the process of finding the most favorable solution or outcome, particularly within the constraints of certain conditions. In mathematical terms, optimization often involves maximizing or minimizing a given function. In this exercise, we aim to maximize the sum \(x_1 + x_2 + x_3 + x_4\). In optimization problems, we define an objective function to describe what needs to be optimized. Here, our objective function is the sum of four variables. This problem involves ensuring that any solution satisfies a specific condition: the sum of the squares of the variables must equal 16. The presence of this constraint classifies it as a constrained optimization problem. To find the solution, advanced techniques like Lagrange multipliers are necessary. These techniques enable us to handle the balance between maximizing our objective function and adhering to the constraint.

Mathematical Constraints

Mathematical constraints are specific conditions or limits that solutions to an optimization problem must satisfy. In the given exercise, the constraint is that the sum of the squares of our four variables \(x_1, x_2, x_3,\) and \(x_4\) should be equal to 16. Constraints can often be equality constraints (like in this problem) or inequality constraints (where the values might need to be less than or greater than a certain number). Constraints play a significant role in defining the feasible region of a problem, which refers to the set of all possible solutions that satisfy the constraints. Handling these constraints is essential, as it determines the feasible solutions. Techniques such as Lagrange multipliers allow us to incorporate these conditions directly into our optimization framework. This is crucial for ensuring that our solution not only maximizes or minimizes our objective but also remains valid within the problem's requirements.

Partial Derivatives

Partial derivatives are used to measure how a function changes as its variables change, one at a time. They are essential in multivariable calculus and play a crucial role in optimization, especially when dealing with more than one variable. In the Lagrange function we write for this problem, partial derivatives help us identify extreme values by showing how small changes in each variable affect the overall function. The solution involves calculating the partial derivatives of our Lagrange function with respect to each variable, including our Lagrange multiplier \(\lambda\). For example:

• \(\frac{\partial L}{\partial x_1} = 1 - 2\lambda x_1 = 0\) helps determine how \(x_1\) influences the function.
• \(\frac{\partial L}{\partial \lambda} = x_1^2 + x_2^2 + x_3^2 + x_4^2 - 16 = 0\) reflects our constraint directly.

By setting each of these partial derivatives to zero, we can solve for the variables, leading us to the solution that both optimizes the objective function and satisfies the constraint.