Let `f(x)=abs(2x^2+5abs(x)-3)` If m is the number of points where f(x) is discontinuous and m is the number of points where f(x) is non-differentiable then value of m+n is

Options

A. 2
B. 0
C. 3
D. 4

Answer

To solve the problem, we need to analyze the function \( f(x) = |2x^2 + 5|x| - 3| \) for points of discontinuity and non-differentiability.

### Step 1: Identify points where the expression inside the absolute value changes sign.
The function has two absolute value components: \( |x| \) and \( |2x^2 + 5|x| - 3| \).

1. **For \( |x| \)**:
- The expression \( |x| \) changes at \( x = 0 \).

2. **For \( 2x^2 + 5|x| - 3 \)**:
- We need to find when \( 2x^2 + 5|x| - 3 = 0 \).
- For \( x \geq 0 \): \( |x| = x \)
\[ 2x^2 + 5x - 3 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm 7}{4} \]
This gives us \( x = \frac{1}{2} \) and \( x = -3 \) (only \( x = \frac{1}{2} \) is valid for \( x \geq 0 \)).

- For \( x < 0 \): \( |x| = -x \)
\[ 2x^2 - 5x - 3 = 0 \]
Again using the quadratic formula:
\[ x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm 7}{4} \]
This gives us \( x = 3 \) and \( x = -\frac{1}{2} \) (only \( x = -\frac{1}{2} \) is valid for \( x < 0 \)).

### Step 2: List all critical points.
The critical points where \( f(x) \) may be non-differentiable or discontinuous are:
- \( x = 0 \)
- \( x = \frac{1}{2} \)
- \( x = -\frac{1}{2} \)

### Step 3: Check for discontinuities.
Since \( f(x) \) is composed of polynomials and absolute values, it is continuous everywhere. Therefore, the number of points of discontinuity \( m = 0 \).

### Step 4: Check for non-differentiability.
At the points \( x = 0 \), \( x = \frac{1}{2} \), and \( x = -\frac{1}{2} \), the function changes its slope due to the absolute value. Therefore, these points are non-differentiable.

Thus, the number of points where \( f(x) \) is non-differentiable \( n = 3 \).

### Final Calculation:
Now, we need to find \( m + n \):
\[ m + n = 0 + 3 = 3 \]

### Conclusion:
The value of \( m + n \) is \( \boxed{3} \).