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Let f(x)=([a]^(2)-5[a]+4)x^(3)-(6{a}^(2)...

Let `f(x)=([a]^(2)-5[a]+4)x^(3)-(6{a}^(2)-5{a}+1)x-(tanx)xx"sgn " x ` be an even function for all ` x in R.` Then the sum of all possible values of a is (where `[*]` and `{*}` denote greatest integer function and fractional part function, respectively)

A

`(17)/(6)`

B

`(53)/(6)`

C

`(31)/(3)`

D

`(35)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
D
`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)`
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