Functions
Functions
Allen Star Batch
Grade 12
Question:
Let $f(x) = ([a]^2 - 5[a] + a)x^3 - (8[a]^2 - 5[a] + 1)x - (\tan x)\operatorname{sgn}x$, be an even function for all $x \in \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$, then sum of all possible values of $a$ is: (where $[.]$ and $\{.\}$ denotes greatest integer function and fractional part functions, respectively)
$\frac{17}{6}$
$\frac{53}{6}$
$\frac{35}{3}$
None of these
Step-by-Step Solution
Key Concept: Use the even function property to eliminate odd terms and apply functional equation constraints to isolate conditions on $[a]$ and $|a|$ separately.
Since $f(x)$ is even, $f(x) = f(-x)$ for all $x \in \mathbb{R} \setminus \{(2n+1)\frac{\pi}{2}\}$. Expanding both sides of the functional equation and using evenness eliminates odd-function terms. By comparing coefficients and using the condition that $\sgn(x) + \sgn(-x) = 0$ and $\tan x + \tan(-x) = 0$ at allowed points, we obtain $[a]^2 - 5[a] + 4 = 0$ and $6|a|^2 - 5|a| + 1 = 0$. Solving: $([a]-1)([a]-4) = 0$ and $(3|a|-1)(2|a|-1) = 0$, giving $a \in \{1, 4, \frac{1}{3}, \frac{1}{2}\}$.
Correct Answer: 3