Question:
The maximum of the function \( f(x, y, z) = xyz \) subject to the constraints
\[
xy + yz + zx = 12, \quad x>0, y>0, z>0,
\]
is equal to (round off to TWO decimal places):
The maximum of the function \( f(x, y, z) = xyz \) subject to the constraints
\[
xy + yz + zx = 12, \quad x>0, y>0, z>0,
\]
is equal to (round off to TWO decimal places):
Show Hint
For constrained optimization, use symmetry and constraints to simplify calculations.
Updated On: Feb 1, 2025
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Solution and Explanation
Step 1: Applying Lagrange multipliers.
Define the Lagrangian:
\[
\mathcal{L}(x, y, z, \lambda) = xyz + \lambda(12 - xy - yz - zx).
\]
Step 2: Solving the system of equations.
Taking partial derivatives and solving the resulting system yields the critical points. Substituting into the constraint:
\[
x = y = z = 2.
\]
Step 3: Evaluating \( f(x, y, z) \).
\[
f(2, 2, 2) = 2 \cdot 2 \cdot 2 = 8.
\]
Step 4: Conclusion.
The maximum value is \( {8} \).
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