The function f(x, y, z) = xyz takes its absolute maximum value of 1/√27 at the six corner points of the ellipsoid where two coordinates have the same sign and the third has the opposite sign. It takes its absolute minimum value of -1/√27 at the six remaining corner points where all three coordinates have the same sign.
Finding the absolute maximum and minimum values of f(x, y, z) using Lagrange Multipliers:
Set up the Lagrange equation: Introduce a Lagrange multiplier (λ) and form the function L(x, y, z, λ):
L(x, y, z, λ) = xyz + λ(x² + 4y² + 4z² - 4)
Find the gradients: Calculate the gradients of L with respect to x, y, z, and λ:
∇L = (yz + 2xλ, xz + 8yλ, xy + 8zλ, x² + 4y² + 4z² - 4)
Set the gradients equal to zero: Set each component of the gradient to zero and treat λ as an unknown variable:
yz + 2xλ = 0
xz + 8yλ = 0
xy + 8zλ = 0
x² + 4y² + 4z² - 4 = 0
Solve the system of equations: This system of equations has four unknowns (x, y, z, λ). However, we can eliminate λ from three of the equations:
Divide the first equation by y: z + 2(x/y)λ = 0
Divide the second equation by z: x + 8(y/z)λ = 0
Divide the third equation by x: y + 8(z/x)λ = 0
Now, we see that the ratios x/y, y/z, and z/x are all equal to -λ/2. Substituting this into the constraint equation:
x² + 4y² + 4z² = 4
x² + 4(-λ/2)² + 4(-λ/2)² = 4
x² + 4λ² = 4
This reduces to a quadratic equation in λ: λ² + λ - 1 = 0. Factor and solve:
(λ + 1)(λ - 1) = 0
λ = -1 or λ = 1
Find the critical points: Substitute each value of λ back into the equations to find the corresponding critical points (x, y, z):
For λ = -1: x/y = y/z = z/x = 1/2
x² + y² + z² = 1
(x, y, z) = (±1/√3, ±1/√3, ±1/√3)
For λ = 1: x/y = y/z = z/x = -1/2
x² + y² + z² = 1
(x, y, z) = (±1/√3, ±1/√3, ±1/√3)
Evaluate f(x, y, z) at the critical points:
f(1/√3, 1/√3, 1/√3) = 1/√27
f(-1/√3, -1/√3, -1/√3) = -1/√27
f(1/√3, 1/√3, -1/√3) = -1/√27
f(-1/√3, -1/√3, 1/√3) = -1/√27
Compare with boundary points: The ellipsoid is bounded, so we need to compare the critical points with the function's behavior on the boundary.
However, due to the symmetry of the problem and the ellipsoid, the critical points already lie on the boundary.
Conclusion:
Absolute maximum: f(1/√3, 1/√3, 1/√3) = f(-1/√3, -1/√3, -1/√3) = 1/√27 is the absolute maximum value of f.
Absolute minimum: f(1/√3, 1/√3, -1/√3) = f(-1/√3, -1/√3, 1/√3) = -1/√27 is the absolute minimum value of f.
Therefore, the function f(x, y, z) = xyz takes its absolute maximum value of 1/√27 at the six corner points of the ellipsoid where two coordinates have the same sign and the third has the opposite sign.