Question

Let\(f(x) = |2x^2 + 5|x - 3|, x \in \mathbb{R}\). If \(m\) and \(n\) denote the number of points were \(f\)is not continuous and not differentiable respectively, then\(m + n\)is equal to:

• A. 5
• B. 2
• C. 0
• D. 3

Answer 1

The Correct Option is D

We analyze the function \( f(x) = |2x^2 + 5|x| - 3| \) in two steps: checking continuity and differentiability.

Step 1: Continuity

The function \( f(x) \) is a composition of absolute values and polynomials, which are continuous everywhere. Hence, \( f(x) \) is continuous for all \( x \in \mathbb{R} \).
\[ m = 0 \quad (\text{Number of points where } f(x) \text{ is not continuous}) \]

Step 2: Differentiability
 

The function \( f(x) \) involves absolute values, which may cause non-differentiability at specific points:

1. First, the outermost absolute value \( |2x^2 + 5|x| - 3| \) is non-differentiable where \( 2x^2 + 5|x| - 3 = 0 \). Solving: \[ 2x^2 + 5|x| - 3 = 0 \implies \text{Critical points are } x = -\frac{3}{2}, 0, \frac{3}{2}. \]
2. Additionally, the inner term \( |x| \) is non-differentiable at \( x = 0 \).

Hence, the total number of points of non-differentiability is:
\[ n = 3 \quad (\text{at } x = -\frac{3}{2}, 0, \frac{3}{2}). \]

Final Calculation
\[ m + n = 0 + 3 = 3. \]