# Finding Extremes of x^3+y^3+z^3+xyz for Real Numbers with Constraints

• MHB
• anemone
In summary, the purpose of finding extremes of x^3+y^3+z^3+xyz for real numbers with constraints is to determine the maximum and minimum possible values within a specific range. This can be achieved by considering common constraints such as domain and boundary constraints. The process is different from finding the global maximum or minimum, which involves finding the maximum or minimum value over the entire domain. Various methods and formulas can be used to find these extremes, including the method of Lagrange multipliers and calculus techniques. Finding the extremes of this expression has practical applications in fields such as economics, engineering, and physics. It can help optimize production levels, design structures, and understand the behavior of physical systems.
anemone
Gold Member
MHB
POTW Director
Here is this week's POTW:

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Real numbers $x,\,y$ and $z$ satisfy $x+y+z=4$ and $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{3}$. Find the largest and smallest possible value of the expression $x^3+y^3+z^3+xyz$.

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No one answered last week's POTW. (Sadface) However, you can find the suggested solution below:
Note that $(x+y+z)^3=x^3+y^3+z^3+3(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2)+6xyz$ while $3(x+y+z)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)xyz=3(x+y+z)(xy+xz+yz)=3(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2)+9xyz$. Thus,
$(x+y+z)^3-3(x+y+z)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)xyz=x^3+y^3+z^3-3xyz$.

By assumptions, $x+y+z-4$ and $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{3}$. Hence $63-4xyz=x^3+y^3+z^3-3xyz$, implying $x^3+y^3+z^3+xyz=64$.

Consequently, the expression $x^3+y^3+z^3+xyz$ has only one value 64.

An example of numbers satisfying the conditions is $x=1,\,y=\dfrac{3-3\sqrt{3}}{2}$ and $z=\dfrac{3+3\sqrt{3}}{2}$. Then $\dfrac{1}{x}=1,\,\dfrac{1}{y}=\dfrac{-1-\sqrt{3}}{3}$ and $\dfrac{1}{z}=\dfrac{-1+\sqrt{3}}{3}$.

## 1. What is the purpose of finding extremes of x^3+y^3+z^3+xyz for real numbers with constraints?

The purpose of finding extremes of x^3+y^3+z^3+xyz for real numbers with constraints is to determine the maximum and minimum values of the given expression under certain conditions or limitations. This can help in solving optimization problems and understanding the behavior of the expression.

## 2. What are the constraints that need to be considered when finding the extremes of x^3+y^3+z^3+xyz?

The constraints that need to be considered are the values of x, y, and z, which must be real numbers. Additionally, there may be other constraints such as inequalities or specific ranges for the variables.

## 3. How do you find the maximum and minimum values of x^3+y^3+z^3+xyz for real numbers with constraints?

To find the maximum and minimum values, you can use methods such as differentiation, substitution, or graphing. By taking the derivative of the expression and setting it equal to zero, you can find critical points and determine if they are maximum or minimum values. Substitution can also be used to simplify the expression and make it easier to analyze. Graphing the expression can also give a visual representation of the maximum and minimum values.

## 4. Can there be multiple maximum or minimum values for x^3+y^3+z^3+xyz?

Yes, there can be multiple maximum or minimum values for x^3+y^3+z^3+xyz. This can occur when there are multiple critical points that satisfy the constraints and have the same maximum or minimum value. In some cases, there may also be a range of values that can be considered as the maximum or minimum.

## 5. How can finding the extremes of x^3+y^3+z^3+xyz be applied in real-world situations?

Finding the extremes of x^3+y^3+z^3+xyz can be applied in various fields such as economics, physics, and engineering. For example, in economics, it can be used to determine the maximum profit or minimum cost for a business under certain constraints. In physics, it can help in understanding the behavior of a system with multiple variables. In engineering, it can aid in optimizing designs and finding the most efficient solution.

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