Question

Let $f\left(x\right)=x^2-6x+5$ and $m$ is the number of points of non derivability of $y=|f\left(|x|\right)|$. If $|f\left(|x|\right)|=k,k\in R$ has atleast $m$ distinct solution(s), then the number of integral values of $k$ is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is C $4$

Given:
$f\left(x\right)=x^2-6x+5=(x-1)(x-5)$ is
Vertex of the parabola
$=(-\frac{b}{2a},-\frac{D}{4a})=(3,4)$
The graph of $y=|f\left(|x|\right)|=|(|x|-1)(|x|-5)|$ is,


Clearly $y=|f\left(|x|\right)|$ is non derivable at $5$ points,
For $|f\left(|x|\right)|=k$ to have atleast $5$ solution,
$0<k\le 4$

Hence, the number of integral values of $k$ is $4$.