Let⁡$$f(x)=$$[a]$$2$$−$$5$$[a]$$+4x3$$−$$6{a}2$$−$$5{a}+1x$$−$$($$$$tan$$⁡$$x)$$×sgn⁡x 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

Let⁡$$f(x)=$$[a]$$2$$−$$5$$[a]$$+4x3$$−$$6{a}2$$−$$5{a}+1x$$−$$($$$$tan$$⁡$$x)$$×sgn⁡x 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$$x-tanx$$· Sgnx ,  where α$$=a$$ $$2-5a+4$$ ,β $$=$$ $$6a2-5a+1$$ since f is even function ⇒$$f-x=fx$$ ⇒ $$-$$α$$x3+$$β$$x-tanx$$ ·$$Sgnx=$$α$$x3-$$β$$x-tanx$$ ·Sgnx ⇒ $$2$$α$$x3-$$β$$x=0$$ for all x ∈R⇒α $$=0$$ and β$$=0$$   Identity in x therefore $$a2-5a+4=0$$  and $$6a2-5a+1=0$$ ⇒          $$a=1$$ or a $$=4$$    and  $$a=13$$ or $$12$$     ⇒  a $$=$$ $$1+13$$ ,$$1+12$$,$$4+13$$,$$4+12$$ ⇒  sum of the values of a $$=353$$

Let⁡f(x)=[a]2−5[a]+4x3−6{a}2−5{a}+1x−(tan⁡x)×sgn⁡x 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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