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)=ax^(3)-beta x -(tanx) sgn x`
`f(-x)=f(x)`
or ` -alpha x^(3)+beta x -tan x sgn x =alpha x^(3)-beta x -(tan x ) (sgn x)`
or `2(-alpha x^(2)-beta )x=0 AA x in R`
or `alpha=0 " and " beta =0`
` :. [a]^(2)-5[a]+4=0 " and "6{a}^(2) -5{a} +1=0`
or ` [x] =1,4 " and " {x} =(1)/(2),(1)/(3)`
` :. a=1+(1)/(3),1+(1)/(2),4+(1)/(3),4+(1)/(2)`
Sum of values of `a=(35)/(3)`