Am-gm Inequality

The arithmetic mean–geometric mean inequality states that for positive numbers the arithmetic mean is at least the geometric mean, with equality when all numbers are equal, a workhorse of JEE Advanced optimisation. Applying it to a, b, c gives (a + b + c)/3 ≥ (abc)^{1/3} = 27^{1/3} = 3. Therefore a + b + c ≥ 9, and equality holds when a = b = c = 3, which indeed satisfies the product constraint 3·3·3 = 27. Option 27 confuses the sum with the product. Option 6 would require a geometric mean of 2, contradicting the cube root of 27. Option 3 reports the geometric mean instead of the minimised sum. Hence the minimum is 9. It is worth stressing that AM-GM gives a lower bound for the sum precisely because the product is held fixed; if instead the sum were fixed, the same inequality would bound the product from above, illustrating the duality JEE Advanced often exploits in constrained optimisation. Plausibility check: testing an unequal triple like a = 1, b = 1, c = 27 gives a sum of 29, far above 9, confirming that the symmetric equal case indeed produces the smallest sum.