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Bipin Dubey (13679)

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Split a+b+c in a/2+a/2+b/3+b/3+b/3+c/4+c/4+c/4+c/4

Now using AM>=GM

(a+b+c)/9 >= (a/2xa/2xb/3xb/3xb/3xc/4xc/4xc/4xc/4)^1/9

a²b³c⁴ <= (18/9)⁹2²3³4⁴

a²b³c⁴ <= 4²6³8⁴

Hence maximum value of a²b³c⁴ is 4²6³8⁴

In such type of problems greatest value can be found by taking ratios of a,b,c powers.

a:b:c: = 2:3:4
a+b+c=18
a=4 b=6 c=8

a²b³c⁴ = 4²6³8⁴ (maximum value)

Bipin Kumar Dubey Chemical Dept. IIT Kharagpur

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