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

Let `f(x)=([a]^2-5[a]+4)x^3-(6{a}^2-5{a}+1)x-(tanx)xsgnx`be an even fu

1.	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`A. `(17)/(6)`B. `(53)/(6)`C. `(31)/(3)`D. `(35)/(3)`
Answer» Correct Answer - 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)`