Question

Consider the function \[ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to(-\infty,\infty) \] defined by \[ f(x)=\left(|x|+|x-1|\right)\sin x+[x\sin x] \] where \([x\sin x]\) denotes the greatest integer less than or equal to \(x\sin x\). Let \(\alpha\) be the total number of points in the interval \[ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \] at which \(f\) is NOT continuous, and let \(\beta\) be the total number of points in the interval \[ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \] at which \(f\) is NOT differentiable. Then the value of \[ \alpha+\beta \] is ________.

Hint:

Greatest integer functions are discontinuous whenever the inside expression crosses an integer value.

Answer 1

Correct Answer: 3

Step 1: Understanding the Concept:
We analyze the function in two parts: \( g(x) = (|x| + |x-1|) \sin x \) and \( h(x) = [x \sin x] \). Continuity and differentiability depend on the behavior of the absolute values and the jump points of the greatest integer function.
Step 3: Detailed Explanation:
1. Analysis of \( g(x) = (|x| + |x-1|) \sin x \):
The term is continuous everywhere.
At \( x=0 \): \( g(x) \) behaves like \( (x+1)x = x^2+x \) for \( x>0 \) and \( (-x+1)x = -x^2+x \) for \( x<0 \).
The derivative from right is 1, and from left is 1. Thus, it is differentiable at \( x=0 \). (Basically \( |x| \sin x \) is differentiable because \( \sin 0 = 0 \)).
At \( x=1 \): \( |x-1| \) has a sharp corner. Since \( \sin 1 \neq 0 \), \( g(x) \) is non-differentiable at \( x=1 \).
2. Analysis of \( h(x) = [x \sin x] \):
Let \( p(x) = x \sin x \). In the range \( (-\pi/2, \pi/2) \), \( p(x) \) is even and increases from \( p(0) = 0 \) to \( p(\pi/2) = \pi/2 \approx 1.57 \).
The greatest integer function \( [p(x)] \) jumps where \( p(x) \) takes integer values.
In our range, \( p(x) \) reaches \( 1 \). Let \( p(x_0) = 1 \). Since \( p(x) \) is even, \( p(-x_0) = 1 \) as well.
At \( x=0 \), \( p(0)=0 \). Near 0, \( p(x) \) is small and positive, so \( [p(x)] = 0 \). Continuous and differentiable at \( x=0 \).
At \( x = \pm x_0 \), the function \( h(x) \) jumps from 0 to 1 (or vice versa). These are points of discontinuity and non-differentiability.
3. Conclusion for \( \alpha \) (Discontinuity):
\( f(x) \) is discontinuous only where \( h(x) \) is discontinuous.
Points are \( x = x_0 \) and \( x = -x_0 \). So \( \alpha = 2 \).
4. Conclusion for \( \beta \) (Non-differentiability):
Points of non-differentiability:
- Discontinuity points \( \pm x_0 \) (Total 2).
- Sharp point of \( |x-1| \) at \( x=1 \) (Total 1).
So \( \beta = 2 + 1 = 3 \).
5. Final result:
\( \alpha + \beta = 2 + 3 = 5 \).
Step 4: Final Answer:
The sum \( \alpha + \beta \) is 5.