Question

Bestimmen Sie das maximale Produkt der 3 nichtnegativen Zahlen \(x, y\) und \(z\), deren Summe gleich 105 ist.

Short answer

The maximum product is 42875, achieved when each number is 35.

Step-by-step solution

Define the problem

We need to find the maximum product of three non-negative numbers, \( x, y, \) and \( z \), such that their sum is 105. Thus, we are aiming to maximize the function \( P(x, y, z) = x \cdot y \cdot z \) given the constraint \( x + y + z = 105 \).

Use symmetry to simplify

Since the sum \( x + y + z = 105 \) is symmetric in \( x, y, \) and \( z \), it is logical to consider equalizing them. Assume \( x = y = z \) to find a point to evaluate. Thus \( 3x = 105 \), giving \( x = 35 \).

Calculate potential maximum

Substitute \( x = y = z = 35 \) into the product function: \[P(35, 35, 35) = 35 \times 35 \times 35 = 42875.\]

Verify if the product can be increased

Consider what happens when the values slightly deviate from equality. Suppose \( x = 35 + a, y = 35, z = 35 - a \). The constraint remains satisfied \( x + y + z = 105 \). Substitute these into the product function: \[P(x, y, z) = (35 + a) \times 35 \times (35 - a).\]Calculate the derivative with respect to \( a \) and find that it leads to the optimum at \( a = 0 \), confirming our previous value.

Key concepts

Maximization of Product

When looking to maximize the product of three numbers, you want to find those values that, when multiplied, give the largest possible number. In our exercise, we’re asked to take three non-negative numbers—let’s call them \( x, y, \) and \( z \)—and determine their maximum product given that their sum is 105. The function we are maximizing here is \( P(x, y, z) = x \cdot y \cdot z \). Our task is to figure out the specific values of \( x, y, \) and \( z \) that will maximize \( P \), adhering to the constraint \( x + y + z = 105 \).
As a general approach, balancing values can often yield the maximum product. This is intuitively understood when numbers are close to each other, multiplying them can lead to larger results. However, let's explore how symmetry in equations plays into this.

Lagrange Multipliers

In optimization problems involving constraints, Lagrange multipliers offer a powerful method. They help us find local maxima or minima of a function subject to equality constraints. In our case, we want to maximize \( P(x, y, z) = x \cdot y \cdot z \) with the constraint \( x + y + z = 105 \).
This method involves introducing a new variable, a Lagrange multiplier, typically denoted as \( \lambda \), which helps to connect the original function with the constraint.
By setting up and solving the resulting system of equations—formed by the function and the constraint—we can determine critical points that inform where the maximum product might occur. However, due to symmetry present in our specific problem, equalizing \( x, y, \) and \( z \) provided a simpler and immediate solution. This balanced approach often surfaces in symmetric problems.

Symmetry in Equations

Symmetry is an essential aspect in optimization problems, especially when dealing with products and sums. Notice that in the condition \( x + y + z = 105 \), the terms have been structured symmetrically. This hints that if \( x, y, \) and \( z \) are close or equal, the product \( x \cdot y \cdot z \) might reach its peak value.
In this particular problem, symmetry allows us to simplify the scenario by considering \( x = y = z \). Doing so, we find that when these numbers are equal, \( x = y = z = 35 \), their product \( P = 35 \cdot 35 \cdot 35 = 42875 \) is maximized. The symmetry not only reduces the computations but also helps us confirm through testing minor deviations that this is indeed the best setup.

Non-negative Numbers Constraint

Working with non-negative numbers adds another level of focus in our problem. This constraint ensures that values of \( x, y, \) and \( z \) are never below zero.
When applying this constraint alongside our condition \( x + y + z = 105 \), we effectively limit the possible range for \( x, y, \) and \( z \). Therefore, values need to be non-negative to maintain mathematical validity for real-world scenarios.
This restriction plays a crucial role in informing us that some solutions, such as negative or overly large positive numbers, are not feasible. By abiding by this, we are assured that our solution—having \( x = y = z = 35 \)—respects all necessary boundaries, leading us straight to the optimal product.