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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

`a^4b^2c^2` is greatest then `a=(1)/(4)`

B

`a^4b^2c^2` is greatest then `b=(1)/(4)`

C

`a^4b^2c^2` is greatest then `c=(1)/(12)`

D

greatest value of `a^4b^2c^2 is (1)/(9.4^8)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
A.M. `ge` G.M
`implies ((2a)/(4) + (2a)/(4) + (2a)/(4) + (2a)/(4) + (b)/(2) + (b)/(2) + (3c)/(2) + (3c)/(2))/(8)`
`ge .^(8)sqrt((2a)/(4).(2a)/(4).(2a)/(4).(2a)/(4).(b)/(2).(b)/(2).(3c)/(2).(3c)/(2))`
`implies (2a + b + 3c)/(8) ge ((3^(2))/(2^(8)) a^(4) b^(2) c^(2))^(1/8)`
The greatest value takes place when A.M =G.M ad
`(2a)/(4) = (b)/(2) = (3c)/(2)`
`implies a = b = 3c = K`
Now, `2a + b + 3c = 1`
`implies 2K + K + K = 1`
`implies K = 1//4`
`implies a = b = 1//4` and `c = 1//12`
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