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:
To address this problem, we analyze \( f(x) = |2x^2 + 5|x - 3| \), a function incorporating absolute value and multiplication. The objective is to determine the count of points where the function exhibits discontinuity and non-differentiability.
Discontinuity Identification:
The function \( f(x) \) may be discontinuous where \( |x - 3| \) changes sign, specifically at \( x = 3 \). Elsewhere, \( f(x) \) remains continuous because \( |2x^2 + 5| \) is a polynomial and thus continuous everywhere. Consequently, \( f(x) \) is continuous except potentially at \( x = 3 \). Thus, \( m = 0 \), as no additional points of discontinuity are found.
Non-Differentiability Identification:
The function \( f(x) = |2x^2 + 5|x - 3| \) includes the term \( |x - 3| \), which is non-differentiable at \( x = 3 \). Potential additional non-differentiability could occur where \( 2x^2 + 5 = 0 \) and \( x - 3 = 0 \) simultaneously.
Solving \( 2x^2 + 5 = 0 \): This equation yields no real roots because \( 2x^2 + 5>0 \) for all real \( x \).
Assessing non-differentiability at \( x = 3 \): The term \( |x - 3| \) introduces non-differentiability at \( x = 3 \). Therefore, \( n = 1 \).
Summation of Discontinuities and Non-Differentiable Points:
The total count is \( m + n = 0 + 1 = 1 \).
The function is continuous at all points except potentially at \( x = 3 \). It is non-differentiable at \( x = 3 \) due to the presence of \( |x-3| \).
Hence, the correct answer is 1, with \( m + n = 1 \).
