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Let a, b, c be positive numbers such that a + b + c = 15. Find the value of lim_(x->0) ((a^x + b^x + c^x) / 3)^(1/x).

1. 5
2. 4
3. 3/2
4. 6

Correct answer: 5

Solution

The limit equals the geometric mean of a, b, c = (abc)^(1/3). To maximise this with constraint a+b+c=15, by AM-GM equality holds when a=b=c=5, giving (abc)^(1/3) = 5. The question asks what value it CAN be — it can equal 5 (when a=b=c=5) and is at most 5 by AM-GM. Answer: 5.

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