# Find Maximum of xyz: Real Solutions

• MHB
• solakis1
In summary, to find the maximum value of xyz when y^2=xz and x+y+z=a, we can use the AM-GM inequality and calculus to find that the maximum value of xyz is 0.
solakis1
Find the maximum of xyz when $y^2=xz$ and $x+y+z=a$

x,y,z are reals

Hello everyone,

I have been working on a problem and I am stuck. I am trying to find the maximum value of the product xyz when y^2=xz and x+y+z=a, where x,y,z are real numbers. I have tried using derivatives and setting up equations, but I am not getting anywhere. Can someone please help me out?

I would approach this problem by first understanding the given constraints and then finding a way to optimize the given equation.

Firstly, we know that y^2=xz and x+y+z=a. This means that we can substitute y^2 for xz in the second equation, giving us y^2+y+z=a. We can then rearrange this equation to get z=a-y-y^2.

Next, we can substitute this value of z into the first equation, giving us y^2=x(a-y-y^2). Simplifying this, we get y^2+xy+y^3=xa.

To find the maximum value of xyz, we can use the AM-GM inequality, which states that the arithmetic mean of a set of numbers is always greater than or equal to the geometric mean of the same set of numbers. In this case, the numbers are y, x, and y^3. Using this inequality, we can write:

(x+y+y^3)/3 ≥ (xy^3)^(1/3)

Simplifying this, we get:

x+y+y^3 ≥ 3xy^(1/3)

Now, using the fact that y^2=xz, we can substitute xz for y^2 in the above equation, giving us:

x+z+y^3 ≥ 3xz^(1/3)

Finally, substituting z=a-y-y^2 and simplifying, we get:

x+a-y-y^2+y^3 ≥ 3(a-y-y^2)^(1/3)

To find the maximum value of xyz, we need to find the maximum value of this expression. To do this, we can use calculus and take the derivative of this expression with respect to y. Setting the derivative equal to 0 and solving for y, we get y=1.

Substituting y=1 back into our original equation, we get x+z=1. Since we want to find the maximum value of xyz, we can set z=0 and x=1, giving us the maximum value of xyz as 0

## 1. What does "Find Maximum of xyz: Real Solutions" mean?

"Find Maximum of xyz: Real Solutions" refers to finding the highest possible value for the product of three variables, x, y, and z, where all three variables are real numbers.

## 2. Why is it important to find the maximum of xyz?

Finding the maximum of xyz can be important in various fields of science, such as economics, physics, and engineering. It can help determine the most efficient or optimal solution to a problem, or identify the highest possible yield or profit.

## 3. How do you find the maximum of xyz?

To find the maximum of xyz, you can use calculus to find the critical points of the function xyz and then evaluate these points to determine the maximum value. Alternatively, you can use algebraic methods such as completing the square or the AM-GM inequality.

## 4. Can there be more than one maximum for xyz?

Yes, there can be more than one maximum for xyz. This can occur when the function has multiple critical points with the same maximum value, or when there is a plateau where the function remains constant at the maximum value for a range of values.

## 5. Are there any limitations to finding the maximum of xyz?

Yes, there are limitations to finding the maximum of xyz. For example, if the function is not continuous or differentiable, the methods used to find the maximum may not be applicable. Additionally, if the function has an infinite number of critical points, it may be difficult to determine the maximum value.

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