Let f(x) = |2 x ^ 2 + 5|x| - 3|, x ∈ R . If m and n denote the...

Given: $f\left(x\right) = \left|2{x}^{2} + 5\left|x\right| - 3\right|$

Now, let $y = \left|2{x}^{2} + 5x - 3\right|$

$⇒ y = \left|2{x}^{2} + 6x - x - 3\right|$

$⇒ y = \left|2x\left(x + 3\right) - \left(x + 3\right)\right|$

$⇒ y = \left|\left(2x - 1\right)\left(x + 3\right)\right|$

Graph of $⇒ y = \left|\left(2x - 1\right)\left(x + 3\right)\right|$ is given below:

Graph of $f\left(x\right) = \left|2{x}^{2} + 5\left|x\right| - 3\right|$ is given below:

So, $f\left(x\right)$ is continuous for $x \in R$ and non-differentiable at $x = \pm \dfrac{1}{2}, 0$

So, number of points of discontinuity$= 0 = m$

Also, number of points of non-discontinuity$= 3 = n$

$⇒ m + n = 3$