If `a,b,c,in R^(+)`, such that `a+b+c=18`, then the maximum value of `a^2,b^3,c^4` is equal to

Options

A. `2^(18)xx3^(2)`
B. `2^(18)xx3^3`
C. `2^19 xx3^2`
D. `2^19xx3^3`

Answer

D

Explanation

`a + b + c = 18`
`implies 2 xx (a)/(2) + 3 xx (b)/(3) + 4 xx (c )/(4) = 18`
Using weighted A.M and G.M inequality, we get
`(2 xx (a)/(2) + 3 xx (b)/(3) + 4 xx (c )/(4))/(9) ge (((a)/(2))^(2) ((b)/(3))^(3) ((c )/(4))^(4))^(1//9)`
or `2^(9) ge (a^(2))/(2^(2)) xx (b^(2))/(3^(3)) xx (c^(4))/(4^(4))`
or ` a^(2) b^(3) c^(4) le 3^(3) xx 2^(19)`