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if 3x+7y+4z=21 when x,y,z are positive real numbers then maximum value of (x 4 )*(y 5 )*(z 3 ) then prove that it is 7 7 * 5 5 * 4 -10 /12

if 3x+7y+4z=21 when x,y,z are positive real numbers then maximum value of (x4)*(y5)*(z3) then prove that it is 
77 * 55 * 4-10/12

Grade:11

1 Answers

Lakshya Goswami
13 Points
6 years ago
A.M>G.M
3x can also be written as 3x/4 + 3x/4 + 3x/4 +3x/4
7y can also be written as 7y/5 + 7y/5 + 7y/5 + 7y/5 + 7y/5
4z can also be written as 4z/3 +4z/3 +4z/3
A.M >G.M
(3x/4 + 3x/4 + 3x/4 +3x/4 + 7y/5 + 7y/5 + 7y/5 + 7y/5 + 7y/5 + 4z/3 +4z/3 +4z/3) / 12
> (3x/4 * 3x/4 * 3x/4 *3x/4 * 7y/5 * 7y/5 * 7y/5 * 7y/5 * 7y/5 * 4z/3 * 4z/3 * 4z/3)1/12
= {(3x +7y + 4z) / 12} > {(3/4)4 * (7/5)5 * (4/3)3 * x4y5z3}1/12
= (21/12)12 > (3/4)4 * (7/5)5 * (4/3)3 * x4y5z3
= { (712 * 312) / (412 * 312) * (44/34) * (55/75) * (33/43) }  > x4y5z3
 
= {(77 * 55 * 4-10) / 12 } > x4y5z3

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