Let `f(x)=([a]^2-5[a]+4)x^3-(6{a}^2-5{a}+1)x-(tanx)xsgnx` be an even function for all `x in Rdot` Then the sum of all possible values of `a` is (where `[dot]a n d{dot}` denote greatest integer function and fractional part function, respectively). `(17)/6` (b) `(53)/6` (c) `(31)/3` (d) `(35)/3`

Options

A. `(17)/(6)`
B. `(53)/(6)`
C. `(31)/(3)`
D. `(35)/(3)`

Answer

D

Explanation

`f(x)=alphax^(3)-betax-(tanx)"sgn "x`
`"since "f(-x)=f(x)`
`rArr" "-alphax^(3)+betax-tanx"sgn "x=alphax^(2)-betax-(tanx)"sgn x"`
`rArr" "2(alphax^(2)-beta)x=0 AA x in R`
`rArr" "alpha=0 and beta=0`
`therefore" "[a]^(2)-5[a]+4=0 and 6{a}^(2)-5{a}+1=0`
`rArr" "([a]-1)([a]-4)=0 and (3{x}-1)(2{x}-1)=0`
`rArr" "[a]=1,4 and {a}=1//3,1//2`
`therefore" "a=1+(1)/(3),1+(1)/(4),4+(1)/(3),4+(1)/(2)`
Sum of values of `a=(35)/(3)`