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Answer :
The maximum value of f(x, y, z) on the given sphere will be 92√2/√47 and the minimum value will be -92√2/√47.
To find the maximum and minimum values of the function f(x, y, z) = 2x - 9y + 3z on the sphere x² + y² + z² = 94, we can use the method of Lagrange multipliers. This method allows us to optimize a function subject to a constraint. The maximum value of f(x, y, z) on the given sphere will be 92√2/√47 and the minimum value will be -92√2/√47.
1. First, let's define the constraint equation:
x² + y² + z² = 94
2. Next, we need to find the gradient vector of both the function f(x, y, z) and the constraint equation. The gradient of f(x, y, z) is given by:
∇f(x, y, z) = <∂f/∂x, ∂f/∂y, ∂f/∂z> = <2, -9, 3>
The gradient of the constraint equation is given by:
∇g(x, y, z) = <∂g/∂x, ∂g/∂y, ∂g/∂z> = <2x, 2y, 2z>
3. We can now set up the equations for Lagrange multipliers:
∇f(x, y, z) = λ∇g(x, y, z)
<2, -9, 3> = λ<2x, 2y, 2z>
4. Equating the corresponding components, we have the following system of equations:
2 = 2λx
-9 = 2λy
3 = 2λz
x² + y² + z² = 94
5. Solving the first three equations for x, y, and z in terms of λ, we get:
x = 1/λ
y = -9/2λ
z = 3/2λ
6. Substituting these values into the constraint equation, we have:
(1/λ)² + (-9/2λ)² + (3/2λ)² = 94
7. Simplifying the equation, we get:
1/λ² + 81/4λ² + 9/4λ² = 94
1/λ² + 90/4λ² = 94
1/λ² + 45/2λ² = 94
2/2λ² + 45/2λ² = 94
(2 + 45)/2λ² = 94
47/2λ² = 94
λ² = 47/2
λ = ±√(47/2)
8. Now we can substitute the values of λ back into x, y, and z to find the corresponding points (x, y, z) on the sphere.
9. For λ = √(47/2), we have:
x = 1/(√(47/2)) = √2/√47
y = -9/(2√(47/2)) = -9√2/√47
z = 3/(2√(47/2)) = 3√2/√47
For λ = -√(47/2), we have:
x = 1/(-√(47/2)) = -√2/√47
y = -9/(2(-√(47/2))) = 9√2/√47
z = 3/(2(-√(47/2))) = -3√2/√47
10. We can now evaluate the function f(x, y, z) at these points to find the maximum and minimum values.
For λ = √(47/2):
f(x, y, z) = 2(√2/√47) - 9(-9√2/√47) + 3(3√2/√47) = 2√2/√47 + 81√2/√47 + 9√2/√47 = 92√2/√47
For λ = -√(47/2):
f(x, y, z) = 2(-√2/√47) - 9(9√2/√47) + 3(-3√2/√47) = -2√2/√47 - 81√2/√47 - 9√2/√47 = -92√2/√47
Therefore, the maximum value of f(x, y, z) on the given sphere is 92√2/√47 and the minimum value is -92√2/√47.
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Therefore, the maximum value of f(x, y, z) on the given sphere is 94, and the minimum value is -94. To find the maximum and minimum values of the function f(x, y, z) = 2x – 9y + 3z on the sphere x² + y² + z² = 94, we can use the method of Lagrange multipliers.
First, let's define the Lagrange function:
L(x, y, z, λ) = 2x – 9y + 3z + λ(x² + y² + z² - 94)
Next, we need to find the critical points by taking the partial derivatives of L with respect to x, y, z, and λ, and setting them equal to zero:
∂L/∂x = 2 + 2λx = 0 (1)
∂L/∂y = -9 + 2λy = 0 (2)
∂L/∂z = 3 + 2λz = 0 (3)
∂L/∂λ = x² + y² + z² - 94 = 0 (4)
Solving equations (1), (2), and (3) for x, y, z in terms of λ gives:
x = -1/λ (5)
y = 9/(2λ) (6)
z = -3/(2λ) (7)
Substituting equations (5), (6), and (7) into equation (4) gives:
(-1/λ)² + (9/(2λ))² + (-3/(2λ))² - 94 = 0
Simplifying this equation yields:
1/λ² + 81/(4λ²) + 9/(4λ²) - 94 = 0
4 + 81 + 9 - 376λ² = 0
376λ² = 94
λ² = 94/376
λ² = 1/4
λ = ±1/2
Considering both positive and negative values of λ, we have two sets of solutions:
Set 1: λ = 1/2
Substituting this value of λ into equations (5), (6), and (7) gives:
x = -1/(1/2) = -2
y = 9/(2(1/2)) = 9
z = -3/(2(1/2)) = -3
So one critical point is (-2, 9, -3).
Set 2: λ = -1/2
Substituting this value of λ into equations (5), (6), and (7) gives:
x = -1/(-1/2) = 2
y = 9/(2(-1/2)) = -9
z = -3/(2(-1/2)) = 3
So another critical point is (2, -9, 3).
To determine if these critical points are maximum or minimum, we need to evaluate the function f(x, y, z) at each point.
f(-2, 9, -3) = 2(-2) – 9(9) + 3(-3) = -4 - 81 - 9 = -94
f(2, -9, 3) = 2(2) – 9(-9) + 3(3) = 4 + 81 + 9 = 94
Therefore, the maximum value of f(x, y, z) on the given sphere is 94, and the minimum value is -94.
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