Verify that the sum of three quantities x, y, z

In summary, the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal. Assuming that my understanding of the question is correct, x = y = z is not the maximum because the rate of change of w is not zero. k = 1 and x = 100, y = 0.1, z = 0.1 are not the maximum either because x = y = z does not give w = 3. However, if k = 1 and x = 100, y = 0.1, z = 0.1 then k is still 1 and w = 100.2.
  • #1
Elias Waranoi
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2

Homework Statement


Verify that the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal.

Homework Equations


w = x + y + z
k = x * y * z

The Attempt at a Solution


Assuming that my understanding of the question is correct i.e. that we want the maximum of w and x*y*z has to be equal to a constant k then I'm pretty lost. If they mean the maximum as the largest value of w possible then it's definitely not when x = y = z.

If k = 1 then if x = y = z would give w = 3. But if k = 1 and x = 100, y = 0.1, z = 0.1 then k is still 1 and w = 100.2.

If they mean maximum as in where the rate of change of w is 0 then that doesn't make sense either because even though the rate of change of w is 0 when x = y = z, w is at it's lowest so that would make that point into a minimum.

The only thing I can think of is that they wrote maximum instead of minimum.
 
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  • #2
Haven't you seen the method of Lagrange multipliers?
 
  • #3
Never heard of it. I'm working with the Calculus made easy book and I haven't encountered it there.
 
  • #4
What kind of techniques have you seen/used to solve these types of problems?
 
  • #5
This chapter was the first and only chapter on partial differentation. They showed that to find the maxima and minima of a multivariable function, the derivative of all partial derivatives have to be zero.
 
  • #6
Elias Waranoi said:

Homework Statement


Verify that the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal.

Homework Equations


w = x + y + z
k = x * y * z

The Attempt at a Solution


Assuming that my understanding of the question is correct i.e. that we want the maximum of w and x*y*z has to be equal to a constant k then I'm pretty lost. If they mean the maximum as the largest value of w possible then it's definitely not when x = y = z.

If k = 1 then if x = y = z would give w = 3. But if k = 1 and x = 100, y = 0.1, z = 0.1 then k is still 1 and w = 100.2.

If they mean maximum as in where the rate of change of w is 0 then that doesn't make sense either because even though the rate of change of w is 0 when x = y = z, w is at it's lowest so that would make that point into a minimum.

The only thing I can think of is that they wrote maximum instead of minimum.
You are right. The sum should be minimum.
 
  • #7
Ah, sorry, I missed that, I thought they were asking for a minimum.
 
  • #8
Elias Waranoi said:

Homework Statement


Verify that the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal.

Homework Equations


w = x + y + z
k = x * y * z

The Attempt at a Solution


Assuming that my understanding of the question is correct i.e. that we want the maximum of w and x*y*z has to be equal to a constant k then I'm pretty lost. If they mean the maximum as the largest value of w possible then it's definitely not when x = y = z.

If k = 1 then if x = y = z would give w = 3. But if k = 1 and x = 100, y = 0.1, z = 0.1 then k is still 1 and w = 100.2.

If they mean maximum as in where the rate of change of w is 0 then that doesn't make sense either because even though the rate of change of w is 0 when x = y = z, w is at it's lowest so that would make that point into a minimum.

The only thing I can think of is that they wrote maximum instead of minimum.

The result is true for a minimum sum. There is no maximum sum under those conditions, because we can write the sum as
$$S = x + y + \frac{k}{xy},$$
and so can have ##S \to +\infty## by taking ##x,y \to 0+##.
 
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  • #9
In that case, k = xyz, w = x + y + z = k/(yz) + k/(xz) + k/(xy)
∂w/∂x = -k/(x2z) - k/(x2y) = -y/x - z/x = 0 ∴ y = -z
∂w/∂y = -k/(y2z) - k/(y2x) = -x/y - z/y = 0 ∴ z = -x
∂w/∂z = -k/(z2y) - k/(z2x) = -x/z - y/z = 0 ∴ x = -y

Now this confuses me, because y = -z therefore x = -y = z. Now I have x = z and z = -x? Am I allowed to do this?

Elias Waranoi said:
If k = 1 then if x = y = z would give w = 3
I tried testing with values and w doesn't even seem to be a minimum when x = y = z considering that if x = 1, y = -1, z = -1 then w = -1
 
  • #10
What
Elias Waranoi said:

Homework Statement


Verify that the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal.

Homework Equations


w = x + y + z
k = x * y * z

The Attempt at a Solution


Assuming that my understanding of the question is correct i.e. that we want the maximum of w and x*y*z has to be equal to a constant k then I'm pretty lost. If they mean the maximum as the largest value of w possible then it's definitely not when x = y = z.

If k = 1 then if x = y = z would give w = 3. But if k = 1 and x = 100, y = 0.1, z = 0.1 then k is still 1 and w = 100.2.

If they mean maximum as in where the rate of change of w is 0 then that doesn't make sense either because even though the rate of change of w is 0 when x = y = z, w is at it's lowest so that would make that point into a minimum.

The only thing I can think of is that they wrote maximum instead of minimum.
constraints do you have on x,y,z ? Are they positive?
 
  • #11
Elias Waranoi said:
In that case, k = xyz, w = x + y + z = k/(yz) + k/(xz) + k/(xy)
∂w/∂x = -k/(x2z) - k/(x2y) = -y/x - z/x = 0 ∴ y = -z
∂w/∂y = -k/(y2z) - k/(y2x) = -x/y - z/y = 0 ∴ z = -x
∂w/∂z = -k/(z2y) - k/(z2x) = -x/z - y/z = 0 ∴ x = -y

Now this confuses me, because y = -z therefore x = -y = z. Now I have x = z and z = -x? Am I allowed to do this?I tried testing with values and w doesn't even seem to be a minimum when x = y = z considering that if x = 1, y = -1, z = -1 then w = -1
Elias Waranoi said:
In that case, k = xyz, w = x + y + z = k/(yz) + k/(xz) + k/(xy)
∂w/∂x = -k/(x2z) - k/(x2y) = -y/x - z/x = 0 ∴ y = -z
∂w/∂y = -k/(y2z) - k/(y2x) = -x/y - z/y = 0 ∴ z = -x
∂w/∂z = -k/(z2y) - k/(z2x) = -x/z - y/z = 0 ∴ x = -y

Now this confuses me, because y = -z therefore x = -y = z. Now I have x = z and z = -x? Am I allowed to do this?I tried testing with values and w doesn't even seem to be a minimum when x = y = z considering that if x = 1, y = -1, z = -1 then w = -1

Please use the appropriate reply button that identifies which post you are replying to. I cannot tell what you are commenting on in your response above.

For non-negative values of ##x,y,z## the point ##(x,y,z) = (c,c,c)## (with ##c = k^{1/3}## and ##k > 0##) really is a provable minimum.
However, if you allow for some negative values of the variables, the sum has no finite minimum for the same reason I outlined in #8 above: you can have
$$S = x + y + \frac{k}{xy} \to -\infty$$
by having ##x \to 0+## and ##y \to 0-## (again, when ##k > 0##).
 
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FAQ: Verify that the sum of three quantities x, y, z

1. What does it mean to "verify" the sum of three quantities?

Verifying the sum of three quantities means to check that the sum of these three numbers is correct or accurate. This involves performing calculations or checks to ensure that the sum is equal to the expected result.

2. What are the three quantities x, y, z referring to?

X, y, and z can represent any three numbers or quantities. These could be measurements, values, or variables in a mathematical equation.

3. How can I verify the sum of three quantities without a calculator?

You can verify the sum of three quantities manually by adding the numbers together using basic arithmetic operations such as addition and subtraction. This can also be done by using a pen and paper or mental math techniques.

4. Why is it important to verify the sum of three quantities?

Verifying the sum of three quantities is important to ensure the accuracy and validity of calculations or equations. It helps to detect any errors or mistakes in the calculation process and ensures that the final result is correct.

5. Are there any common mistakes to watch out for when verifying the sum of three quantities?

Yes, some common mistakes to watch out for include accidentally omitting a number, using the wrong mathematical operation, or making a calculation error. It is important to double-check the numbers and calculations to avoid these mistakes.

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