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Question

# If we write 272 as the sum of positive real numbers, so as to maximize their product, then 272 is written as

A
272=2+2+upto 136 terms
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B
272=2.72+2.72+upto 100 terms
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C
272=34+34+upto 8 terms
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D
272=17+17+upto 16 terms
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Solution

## The correct option is B 272=2.72+2.72+⋯upto 100 termsTo maximize the product, all of the numbers should be equal (using A.M-G.M inequality). Let 272 divided into x equal parts. (x∈N) So, we need to maximize y=(272x)x Taking ln to both the sides, ⇒lny=x(ln272−lnx) Differentiate w.r.t x ⇒1y⋅y′=ln272−x×1x−lnx To find critical point, y′=0 ⇒ln272−1−lnx=0 ⇒x=272e⇒x=100 [∵x∈N] Hence, the maximum product occurs when 272=2.72+2.72+⋯upto 100 terms

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