Positive numbers a , b , c satisfy a + b + c = 15 . Which values can lim x → 0 ( 3 a x + b x + c x ) 1/ x take?

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Positive numbers a , b , c satisfy a + b + c = 15 . Which values can lim x → 0 ​ ( 3 a x + b x + c x ​ ) 1/ x take?

TestHub · Accepted Answer

Taking logarithms gives the limit ( ab c ) 1/3 . By AM-GM, ( ab c ) 1/3 ≤ 5 , with every positive value up to 5 attainable under the fixed-sum condition. Thus 5, 4, and 3/2 are possible, but 6 is not.

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