Let `X_(k)` be real number such that `X_(k)gtk^(4)+k^(2)+1` for `1 le k le 2018`. Denot`N =sum_(k=1)^(2018)k`. Consider the following inequalities:
I. `(sum_(k=1)^(2018)kx_(k))^(2)leN(sum_(k=1)^(2018)kx_(k)^(2)) " " II. (sum_(k=1)^(2018)kx_(k))^(2)leN(sum_(k=1)^(2018)k^(2)x_(k)^(2))`

Options

A. both I and II are true
B. I is true and II is false
C. I is false and II is true
D. both I and II are false

Answer

A

Explanation

If `x_(1),x_(2),……………,x_(n)` be n numbers then using cauch-schurz theorem -
`((x_(1)+x_(2)+.....x_(n))/(n))^(2) le ((x_(1)^(2)+x_(2)^(2)+.....x_(n)^(2))/(n))`
Case (i) Consider
`x_(1),x_(2),x_(2),x_(3),x_(3),x_(3)..............underset(2018 "times")(ubrace(x_(2018)+x_(2018)+......x_(2018)))`
Now using Cauchy-schwarz for above number
`((x_(1),x_(2),x_(2)+..............(x_(2018)+x_(2018)+......x_(2018))/(2018 " times"))/(1+2+3......2018))^(2)`
`le(x_(1)^(2),x_(2)^(2),x_(2)^(2)+..............(x_(2018)^(2)+x_(2018)^(2)+......)/(2018 " times"))/(1+2+3......2018)`
`implies((x_(1)+2x_(2)+3x_(3)+...2018x_(2018))/(sum_(k=1)^(2018)k))^(2) le ((x_(1)^(2)+2x_(2)^(2)+3x_(3)^(2)+...2018x_(2018)^(2))/(sum_(k=1)^(2018)k))^(2)`
`(sum_(k=1)^(2018)kx_(k))^(2)lesum_(k=1)^(2018)(sum_(k=1)^(2018)kx_(k^(2)))`
`(sum_(k=1)^(2018)kx_(k))^(2)leN(sum_(k=1)^(2018)kx_(k^(2))^(2))`
Therefore statement 1 is true.
Case (ii) Consider
`x_(1),2x_(2),3x_(3)+....2018x_(2018)`
Now apply chauchy-Schwarz for above number
`((x_(1)+2x_(2)+3x_(3)+.....2018x_(2018))/(2018))^(2) le (x_(1)^(2)+2x_(2)^(2)+.....(2018x_(2018))^(2))/(2018)`
`implies(x_(1)+2x_(2)+.....2018_(x2018))^(2) le2018 (x_(1)^(2)+4x_(2)^(2)+...(2018)^(2)x_(2018)^(2))`
`implies (sum_(k=1)^(2018)kx_(k))^(2)le 2018(sum_(k=1)^(2018)k^(2)x_(k)^(2))`
Since `n=sum_(k=1)^(2018)k=(2018xx2019)/(2)`
`:.(sum_(k=1)^(2018)kx_(k))^(2)` is always less than or equal to 2018 `sum_(k=1)^(2018)k^(2)x_(k)^(2)`
`:.` It will always be less than `n(sum_(k=1)^(2018)k^(2)x_(k)^(2))`