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If a,b,c are positive real numbers and 2...

If `a,b,c` are positive real numbers and `2a+b+3c=1`, then the maximum value of `a^(4)b^(2)c^(2)` is equal to

A

`(1)/(3*4^(8))`

B

`(1)/(9*4^(7))`

C

`(1)/(9*4^(8))`

D

`(1)/(27*4^(8))`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` Consider positive numbers
`(2a)/(4)`, `(2a)/(4)`,`(2a)/(4)`,`(2a)/(4)`,`(b)/(2)`,`(b)/(2)`,`(3c)/(2)`,`(3c)/(2)`
Using `A.M. ge G.M.`
`((2a)/(4)+(2a)/(4)+(2a)/(4)+(2a)/(4)+(b)/(2)+(b)/(2)+(3c)/(2)+(3c)/(2))/(8)`
`ge (((2a)/(4)*(2a)/(4)*(2a)/(4)*(2a)/(4)*(b)/(2)*(b)/(2)*(3c)/(2)*(3c)/(2))/(8))^((1)/(8)`
`implies(2a+b+3c)/(8) ge ((3^(2))/(2^(8))*a^(4)b^(2)c^(2))^((1)/(8))`
`(1)/(8) ge ((3^(2)a^(4)b^(2)c^(2))^((1)/(8)))/(2)`
`impliesa^(4)b^(2)c^(2) le (1)/(4^(8)*3^(2))`
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