We interpret the function as $f(x)=\left|2x^2+5|x|-3\right|, \qquad x\in \mathbb R.$

We need:

1. the number of points where $f$ is not continuous = $m$
2. the number of points where $f$ is not differentiable = $n$
3. then compute $m+n$.

1. Check continuity

The inner expression is $g(x)=2x^2+5|x|-3.$

Now:

• $x^2$ is continuous for all $x$
• $|x|$ is continuous for all $x$
• hence $g(x)$ is continuous for all $x$
• and $f(x)=|g(x)|$ is also continuous for all $x$

Therefore, there are no points of discontinuity.

So, $m=0.$

2. Check differentiability

A modulus function $|g(x)|$ can fail to be differentiable at points where:

1. $g(x)=0$ and $g'(x)\neq 0$, or
2. $g(x)$ itself is not differentiable.

So first study $g(x)=2x^2+5|x|-3.$

2.1 Where is $g(x)$ not differentiable?

Since $|x|$ is not differentiable at $x=0$, $g(x)$ is also not differentiable at $x=0.$

Check whether this causes non-differentiability of $f$: $g(0)=2\cdot 0^2+5|0|-3=-3\neq 0.$

Near $x=0$, $g(x)$ remains negative (by continuity), so $f(x)=|g(x)|=-g(x)$ in a neighborhood of $0$. Since $g$ is not differentiable at $0$, $f$ is also not differentiable at $x=0.$

So one non-differentiable point is $x=0$.

2.2 Find points where $g(x)=0$

Solve $2x^2+5|x|-3=0.$

Let $t=|x| \ge 0.$ Then $2t^2+5t-3=0.$

Factor: $2t^2+5t-3=(2t-1)(t+3)=0.$

So $t=\frac12 \quad \text{or} \quad t=-3.$

Since $t\ge 0$, only $t=\frac12$ is valid. Hence $|x|=\frac12 \implies x=\pm \frac12.$

2.3 Differentiability of $f$ at $x=\pm \frac12$

For $x\neq 0$, $g$ is differentiable.

For $x>0$, $g(x)=2x^2+5x-3, \qquad g'(x)=4x+5.$ At $x=\frac12$, $g'\left(\frac12\right)=4\cdot \frac12+5=7\neq 0.$ So $|g(x)|$ has a sharp corner at $x=\frac12$. Hence $f$ is not differentiable at $x=\frac12$.

For $x<0$, $g(x)=2x^2-5x-3, \qquad g'(x)=4x-5.$ At $x=-\frac12$, $g'\left(-\frac12\right)=4\cdot \left(-\frac12\right)-5=-7\neq 0.$ So $f$ is not differentiable at $x=-\frac12$.

Thus additional non-differentiable points are $x=\pm \frac12.$

So total number of non-differentiable points is $n=3.$

3. Compute $m+n$

$m+n=0+3=3.$

4. Compare with stored answer

Derived answer: 3

Stored correct answer: B = 3

They match.