quite a good one

Write 271 as the sum of positive real numbers so as to maximize their product

more of calculus than algebra[4]

2 Answers

106

Asish Mahapatra ·2010-03-09 21:47:25

a1+a2+a3+....+an = 271

we need to maximise a1*a2*a3*....*an

from AM-GM

(271)/n â‰¥ (product)^1/n

=> product â‰¤ (271/n)ⁿ

consider f(x) = (271/x)^x

lnf(x) = x*ln(271) - x*lnx
f'(x)/f(x) = ln(271) - 1 - lnx = 0
=> lnx = ln(271/e)

=> x = 271/e = 99.69

so 271 = e*(271/e)

each part is 271/e

correct :)

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