JEEOlympiad
Let a, b, c be positive real numbers with a + b + c = 12. Find the…
Question
Let a, b, c be positive real numbers with a + b + c = 12. Find the maximum possible value of a^2 · b^3 · c.
✓ Verified answer: 6912checked by our engine — not a guess
Step-by-step solution
Apply weighted AM-GM. Split a into two equal parts a/2 and b into three equal parts b/3, giving 6 nonnegative terms summing to a + b + c = 12:
(a/2)+(a/2)+(b/3)+(b/3)+(b/3)+c = 12.
By AM-GM on these 6 terms, their geometric mean is at most 12/6 = 2, so
(a/2)^2 (b/3)^3 c ≤ 2^6 = 64, i.e. a^2 b^3 c ≤ 64 · 4 · 27 = 6912.
Equality requires a/2 = b/3 = c. With a = 2t, b = 3t, c = t and a+b+c = 6t = 12, we get t = 2, so a=4, b=6, c=2, and a^2 b^3 c = 16·216·2 = 6912.
Final answer6912
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