Let $f (x) = ([a]^{2} - 5 [a] + 4) x^{3} - (6 {a}^{2} - 5 {a} + 1) x - (tan x) \times s g n x$ be an even function for all $x \in R$ , then the sum of all possible values of $^{'} 3 a^{'}$
is
(where, [.] and {.} denote the greatest integer function and fractional part functions, respectively)

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Solution

Let $α = [a]^{2} - 5 [a] + 4 a n d β = 6 {a}^{2} - 5 {a} + 1$
$∴ f (x) = α x^{3} - β x - (tan x) \times s g n x$
Also, as $f (x)$ is an even function. $∴ f (- x) = f (x)$
$\Rightarrow - α x^{3} + β x - (tan x) \times s g n x = α x^{3} - β x - (tan x) \times s g n x$
$\Rightarrow 2 (- α x^{2} - β) x = 0 f o r a l l x \in R$
$\Rightarrow α = 0 a n d β = 0$
$∴ [a]^{2} - 5 [a] + 4 = 0 a n d 6 {a}^{2} - 5 {a} + 1 = 0$
$\Rightarrow ([a] - 1) ([a] - 4) = 0 a n d (3 {a} - 1) (2 {a} - 1) = 0$
$\Rightarrow [a] = 1 o r 4 a n d {a} = \frac{1}{3} o r \frac{1}{2}$
$\Rightarrow a = 1 + \frac{1}{3}, 1 + \frac{1}{2}, 4 + \frac{1}{3}, 4 + \frac{1}{2}$
Therefore, sum of all possible values of a is $\frac{35}{3} ∴ 3 a = 35$

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