Question

The number of points of non-differentiability of the function f(x) = [4 + 13sinx] in (0, 2𝜋) is ____.

Answer 1

We are given the function: \[ f(x) = \left\lfloor 4 + 13 \sin x \right\rfloor \] where \(\left\lfloor y \right\rfloor\) denotes the greatest integer less than or equal to \(y\). We are required to find the number of points of non-differentiability of the function \(f(x)\) in the interval \((0, 2\pi)\). 

Step 1: Analyze the behavior of the floor function 
The function \(f(x)\) involves the greatest integer function, which causes non-differentiability at points where the value inside the floor function is an integer. Hence, we need to find the values of \(x\) for which: \[ 4 + 13 \sin x \in \mathbb{Z} \] where \(\mathbb{Z}\) is the set of integers. 

Step 2: Solve for the values of \(x\) \[ 4 + 13 \sin x = n \quad \text{where} \quad n \in \mathbb{Z}. \] This simplifies to: \[ \sin x = \frac{n - 4}{13}. \] For \(x \in (0, 2\pi)\), the sine function takes values in the range \([-1, 1]\), so we must have: \[ -1 \leq \frac{n - 4}{13} \leq 1. \] Multiplying through by 13: \[ -13 \leq n - 4 \leq 13. \] Adding 4 to each part of the inequality: \[ -9 \leq n \leq 17. \] Thus, \(n\) can take integer values from \(-9\) to \(17\), inclusive. That gives us a total of \(17 - (-9) + 1 = 27\) possible integer values for \(n\). 

Step 3: Solve for the corresponding values of \(x\) For each integer value of \(n\), we solve the equation: \[ \sin x = \frac{n - 4}{13}. \] The equation \(\sin x = \frac{n - 4}{13}\) will have two solutions in the interval \((0, 2\pi)\) for each \(n\), unless \(\frac{n - 4}{13} = \pm 1\). However, in this case, we exclude \(n = -9\) and \(n = 17\) because they lead to \(\sin x = -1\) and \(\sin x = 1\), which only have one solution in the interval \((0, 2\pi)\). For the remaining 25 values of \(n\), each will provide two solutions, so the total number of points of non-differentiability is: \[ 2 \times 25 + 2 = 50. \] 

Final Answer: The number of points of non-differentiability of the function \(f(x)\) in \((0, 2\pi)\) is \(50\).