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:
- 5
- 2
- 0
- 3
The Correct Option is D
Approach Solution - 1
To solve this problem, we need to analyze the given function \( f(x) = |2x^2 + 5|x - 3| \), which involves both an absolute value and multiplication operations. We're tasked with determining the number of points where the function is not continuous and differentiating.
- Identify Discontinuities:
- The function \( f(x) \) can be discontinuous at points where \( |x - 3| \) changes behavior, i.e., \( x = 3 \). At other points \( f(x) \) remains continuous, as \( |2x^2 + 5| \) is a polynomial and thus continuous everywhere. Therefore, \( f(x) \) is continuous except possibly at \( x = 3 \). Hence, \( m = 0 \), as no additional points of discontinuity arise.
- Identify Non-Differentiable Points:
- The function \( f(x) = |2x^2 + 5|x - 3| \) involves the product of \( |x - 3| \) which is not differentiable at \( x = 3 \) and potential additional non-differentiability where \( 2x^2 + 5 = 0 \) and \( x - 3 = 0 \).
- Solve \( 2x^2 + 5 = 0 \): This has no real roots since \( 2x^2 + 5 > 0 \) for all real \( x \).
- Check for non-differentiability at \( x = 3 \): The term \( |x - 3| \) contributes non-differentiability at \( x = 3 \). Thus, \( n = 1 \).
- Sum of Discontinuities and Non-Differentiable Points:
- Summing these, we have \( m + n = 0 + 3 = 3 \).
- The function is continuous at all points except possibly at \( x = 3 \). It's non-differentiable at \( x = 3 \) due to the absolute value function \( |x-3| \).
Therefore, the correct answer is 3, where \( m + n = 3 \).
Approach Solution -2
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:
- First, the outermost absolute value \( |2x^2 + 5|x| - 3| \) is non-differentiable where \( 2x^2 + 5|x| - 3 = 0 \). Solving: \[ 2x^2 + 5|x| - 3 = 0 \implies \text{Critical points are } x = -\frac{3}{2}, 0, \frac{3}{2}. \]
- 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
\[ m + n = 0 + 3 = 3. \]
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