JEEOlympiad
Let x, y, z be positive real numbers with x + y + z = 6. Find the…
Question
Let x, y, z be positive real numbers with x + y + z = 6. Find the maximum value of x·y^2·z^3.
✓ Verified answer: 108checked by our engine — not a guess
Step-by-step solution
Apply weighted AM-GM by splitting the constraint into 1+2+3 = 6 equal parts: x + y/2 + y/2 + z/3 + z/3 + z/3 has six terms summing to 6.
By AM-GM their product is maximized when all six are equal: x = y/2 = z/3 = 1, i.e.
x = 1, y = 2, z = 3.
Then x·y^2·z^3 = 1·4·27 = 108.
(Equivalently, Lagrange multipliers give the ratios x:y/2:z/3 = 1:1:1.) Maximum value is 108.
Final answer108
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