If `a-1,a_2, ,a_n` are positive real numbers whose product is a fixed number `c ,` then the minimum value of `a_1+a_2++a_(n-1)+2a_n` is `a_(n-1)+2a_n` is b. `(n+1)c^(1//n)` `2n c^(1//n)` `(n+1)(2c)^(1//n)`

Options

A. `n(2c)^(1//n)`
B. `(n+1)c^(1//n)`
C. `2n c^(1//n)`
D. `(n+1)(2c)^(1//n)`

Answer

A

Explanation

Using `A.M.geG.M.,` we have
`(a_(1)+a_(2)+...+a_(n-1)+2a_(n))/(n)ge(a_(1)a_(2)....2a_(n))^(1n)`
`implies" "(a_(1)+a_(2)+...+a_(n-1)+2a_(n))/(n)ge(2c)^(1//n)" "[becausea_(1)a_(2)....a_(n)=c]`
`implies" "a_(1)+a_(2)+...+a_(n-1)+2a_(n)gtn(2c)^(1//n)`