Question

Find the maximum and minimum values of the function f(x, y, z) = x^2y^2z^2 subject to the constraint x^2 + y^2 + z^2 = 289. Maximum value is _______ , occuring at _________ points (positive integer or "infinitely many"). Minimum value is ________ , occuring at _________ points (positive integer or "infinitely many").

Answer 1

Final Answer;

Find the maximum and minimum values of the function f(x, y, z) = x² y² z² subject to the constraint x² + y² + z² = 289. Maximum value is
`\( (204121)/(4) \)`, occuring at infinitely many points (positive integer or "infinitely many"). Minimum value is 0 , occuring at infinitely many points (positive integer or "infinitely many").

Step-by-step explanation:

The given function is f(x, y, z) = x² y² z² subject to the constraint x² + y² + z² = 289 . To find the maximum and minimum values, we can use the method of Lagrange multipliers.

First, set up the Lagrangian:

`\[ L(x, y, z, \lambda) = x^2y^2z^2 + \lambda(289 - x^2- y^2 - z^2 )`

Now, take the partial derivatives with respect to x, y, z, and
`\( \lambda \)`and set them to zero:

`\[ (\partial L)/(\partial x) = 2xy^2z^2 - 2\lambda x = 0 \]`

`\[ (\partial L)/(\partial y) = 2x^2yz^2 - 2\lambda y = 0 \]`

`\[ (\partial L)/(\partial z) = 2x^2y^2z - 2\lambda z = 0 \]`

`\[ (\partial L)/(\partial \lambda) = 289 - x^2 - y^2 - z^2 = 0 \]`

Solving these equations, we find x = 0, y = 0, z = 0 or
`\( x = \pm \sqrt{(289)/(2)}, y = \pm \sqrt{(289)/(2)}, z = \pm \sqrt{(289)/(2)} \)`. However, the solution x = 0, y = 0, z = 0 does not satisfy the constraint. So, the points where the maximum and minimum occur are
`\( (x, y, z) = (\pm \sqrt{(289)/(2)}, \pm \sqrt{(289)/(2)}, \pm \sqrt{(289)/(2)}) \).`

Now, substitute these points into the function
`( f(x, y, z) = x^2y^2z^2 \))` to find the corresponding values. The maximum value is
`\( (204121)/(4) \)`occurring at infinitely many points, and the minimum value is 0 occurring at infinitely many points.

Answer 2

Maximum value: No maximum value (infinitely many points) and minimum value: 0 (occurs at infinitely many points)

To find the maximum and minimum values of the function
`$f(x, y, z) = x^2y^2z^2$` subject to the constraint
`$x^2 + y^2 + z^2 = 25$`, we can use the method of Lagrange multipliers.

First, let's define the Lagrangian function
`$L(x, y, z, \lambda) = x^2y^2z^2 - \lambda(x^2 + y^2 + z^2 - 25)$`. Here,
`$\lambda$` is the Lagrange multiplier.

To find the critical points, we need to solve the following system of equations:

1.
`$(\partial L)/(\partial x) = 2xy^2z^2 - 2\lambda x = 0$`

2.
`$(\partial L)/(\partial y) = 2x^2yz^2 - 2\lambda y = 0$`

3.
`$(\partial L)/(\partial z) = 2x^2y^2z - 2\lambda z = 0$`

4.
`$(\partial L)/(\partial \lambda) = -(x^2 + y^2 + z^2 - 25) = 0$`

By solving this system of equations, we find the critical points:

1. When x = 0, y = 0, and z = 0, the value of f(x, y, z) is 0. This is the minimum value, which occurs at infinitely many points.

2. Since all three variables can be zero, there is no maximum value for the function f(x, y, z).