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Let a, b, c be positive real numbers with abc = 8. Find the minimum…
Question
Let a, b, c be positive real numbers with abc = 8. Find the minimum possible value of (1 + a)(1 + b)(1 + c).
✓ Verified answer: 27checked by our engine — not a guess
Step-by-step solution
Expand: (1+a)(1+b)(1+c) = 1 + (a+b+c) + (ab+bc+ca) + abc = 9 + (a+b+c) + (ab+bc+ca), using abc = 8.
By AM-GM, a+b+c ≥ 3(abc)^{1/3} = 3·2 = 6, and ab+bc+ca ≥ 3(a^2b^2c^2)^{1/3} = 3·4 = 12.
Both bounds are attained simultaneously when a = b = c = 2 (which satisfies abc = 8). Hence the minimum is 9 + 6 + 12 = 27.
Final answer27
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