Do you really wanna ask for even?

Geometry Level 3

$f(x)=\left( \lfloor a \rfloor^2-5\lfloor a \rfloor +4 \right)x^3-\left( 6\{a\}^2-5\{a\}+1 \right)x -(\tan (x)) \text{sgn}(x)$

Provided that $f(x)$ be an even function for all $x \in \mathbb{R}$ . If sum of all possible values of $a$ is $\dfrac{P}{Q}$ for coprime positive integers $P,Q$ , then find the value of $(P+Q)$ .

Details And Assumptions :

$\lfloor ..\rfloor$ is floor function or greatest integer function.
$\{ .. \}$ is fractional part function.
$\text{sgn}(x)$ is signum function defined as : $\text{sgn}(x)=\begin{cases} 1 \quad , x>0 \\ 0 \quad , x=0 \\ -1 \quad , x<0 \end{cases}$ .

The answer is 38.

1 solution

Prashant Kr
Apr 6, 2015

f(x)= ø $x^{3}$ -ßx-(tanx)sgnx .

f(x)=f(-x).

-ø $x^{3}$ +ßx-(tanx)sgnx=ø $x^{3}$ -ßx-(tanx)sgnx.

Or 2x(-ø $x^{2}$ -ß)=0 for all real x.

So, ø=0 and ß=0 .

So; $[a]^{2}$ -5[a]+4=0 and 6 ${a}^{2}$ -5{a}+1=0 .

(3{x}-1)(2{x}-1)=0 .

So sum of all values of a =35/3
35+3= 38

Do you really wanna ask for even?

The answer is 38.

1 solution

0 pending reports