Find the maximum value of f (x, y, z) = xyz on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 12$ .
A) 8
B) $8 \sqrt { 3 }$
C) 16
D) No maximum, unbounded

Question

Quizplus · Accepted Answer

To find the maximum value of $f(x, y, z) = xyz$ on the sphere $x^2 + y^2 + z^2 = 12$, we can use the method of Lagrange multipliers. However, a more intuitive approach is to recognize that for $x^2 + y^2 + z^2 = 12$, the variables $x, y, z$ are symmetric. The maximum product $xyz$ occurs when $x = y = z$, due to the symmetry and the arithmetic mean-geometric mean inequality. Setting $x = y = z$, we get $3x^2 = 12$, so $x^2 = 4$, and $x = 2$ (considering positive values for maximum product). Thus, the maximum value of $f(x, y, z)$ is $2 \cdot 2 \cdot 2 = 8$.

Find the Maximum Value of F (X, Y, Z) = x2+y2+z2=12x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 12x2+y2+z2=12

Find the Maximum Value of F (X, Y, Z) = $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 12$