$$Letfx=a2-5a+4x3-6a2-5a+1x-tanx(sgnx)$$ be an even function for all x∈R. Then the sum of all possible values of a is (where$$ [.] and $${$$.$$}$$ denote greatest integer function and fractional part function, $$respectively)

Question

$$Letfx=a2-5a+4x3-6a2-5a+1x-tanx(sgnx)$$ be an even function for all x∈R. Then the sum of all possible values of a is (where$$ [.] and $${$$.$$}$$ denote greatest integer function and fractional part function, $$respectively)

Infinity Learn · Accepted Answer

$$fx=$$α$$x3-$$β$$x-tanxsgnx$$ $$f-x=fx$$ or      $$-$$α$$x3+$$β$$x-tanx$$ sgnx $$=$$α$$x3-$$β$$x-tanx(sgnx)$$ or   $$2-$$α$$x2-$$β$$x=0$$∀x∈R or    α$$=0$$    and   β$$=0$$ ∴    $$a2-5a+4=0$$ and  $$6a2-5a+1=0$$ or   $$x=1$$,$$4$$  and $$x=12$$,$$13$$ ∴   $$a=1+13$$,$$1+12$$,$$4+13$$,$$4+12$$ Sum of values of $$a=353$$

$Let f (x) = ({[a]}^{2} - 5 [a] + 4) x^{3} - (6 {\{a\}}^{2} - 5 \{a\} + 1) x - (\tan x) (s g n x)$ 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)

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Letfx=a2-5a+4x3-6a2-5a+1x-tanx(sgnx) be an even function for all x∈R. Then the sum of all possible values of a is (where [.] and {.} denote greatest integer function and fractional part function, respectively)

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$Let f (x) = ({[a]}^{2} - 5 [a] + 4) x^{3} - (6 {\{a\}}^{2} - 5 \{a\} + 1) x - (\tan x) (s g n x)$ 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)