Problem 66 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)=[n+p \sin x], x \in(0, \pi), n \in Z\) and \(p\) is a prime number, where \([\cdot]\) denotes the greatest integer function. Then, the number of points where \(f(x)\) is not differentiable, are (A) 0 (B) \(2(p-1)\) (C) \(2 p-1\) (D) None of these

Step-by-Step Solution

Answer

The function is not differentiable at \(2(p-1)\) points, so option (B) is correct.

Step 1: Understand the Function

The function given is \(f(x) = [n + p \sin x]\). Here, \([ \cdot ]\) represents the greatest integer function, which means it takes the greatest integer less than or equal to the argument. Given that \(p\) is a prime number and \(n\) is an integer, the function will step at integer values.

Step 2: Analyze the Range of \(\sin x\)

For \(x \in (0, \pi)\), the range of \(\sin x\) is \((0, 1)\). Hence, \(p \sin x\) ranges from \(0\) to \(p\). Consequently, \(n + p \sin x\) ranges from \(n\) to \(n + p\). This implies that \(f(x)\) can take integer values from \(n\) to \(n+p-1\).

Step 3: Identification of Points of Non-Differentiability

The greatest integer function \([ \cdot ]\) is not differentiable at integer points. Since \(f(x)\) changes its value at points where \(n+p\sin x\) is an integer, we need to find \(x\) such that \(n + p\sin x = k\) for integers \(k\). This gives \(\sin x = \frac{k-n}{p}\).

Step 4: Determine Valid Integer Values of \(k\)

The values of \(k\) such that \(\sin x\) remains in the interval \((0,1)\) are \(n+1, n+2, \ldots, n+p-1\). This results in \(p-1\) integer values. For each such \(k\), there are two \(x\) values (due to the symmetrical interval \((0,\pi)\)), leading to \(2(p-1)\) total points.

Step 5: Conclusion

The number of points at which \(f(x)\) is not differentiable corresponds to the number of points \(x\) where \( n+p\sin x\) becomes an integer. Calculated as shown, there are a total of \(2(p-1)\) such points.

Key Concepts

Non-Differentiable Points

Differentiability is a fundamental concept in calculus that helps us understand how functions behave. A function is differentiable at a point if it has a well-defined tangent there, meaning the function is smooth and not broken. In the case of functions involving the greatest integer function, also known as the floor function, differentiability may not exist at certain points. The greatest integer function, denoted by \([\cdot]\), implies taking the largest integer less than or equal to a given number. For such functions, differentiability often fails at integer values, where the function has jumps or discontinuities. When given a function like \([n+p \sin x]\), non-differentiable points occur whenever the equation \([n + p \sin x]\) equals an integer. Specifically, if \([k]\) is an integer, the function steps or jumps at these points. Understanding where the jumps happen involves solving the equation \([n+p\sin x]=k\), and identifying the range of \(\sin x\) to ensure it lies between 0 and 1 (as \(x\) is in \((0, \pi)\)).

Prime Numbers in Functions

Prime numbers can significantly alter the behavior of mathematical functions. In our context, let's consider the function \(f(x) = [n + p \sin x]\), where \(p\) is a prime number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.The use of a prime number \(p\) in the function \(p \sin x\) affects how fast or slow the values within the sine range of \(0-1\) scale up to reach the next integer. This stretching or shrinking defines different points where the greatest integer function might jump due to changed integer values of \(n + p\sin x\). As primes do not introduce additional factors or divisors, they create a clean set of values that cause these jumps only at specific prioritized points, leading to the distinct count of non-differentiable points in the function's domain of \((0,\pi)\).

Trigonometric Functions in Calculus

Trigonometric functions like \(\sin x\) play a crucial role in calculus, especially when they are a part of more complex functions. They introduce periodic behavior and are vital in modeling wave-like phenomena.When \(\sin x\) is integrated into a function, it brings along its periodic nature, with values oscillating between -1 and 1. In our particular function, \(f(x) = [n + p \sin x]\), \(\sin x\) is restricted to \((0, \pi)\), where its range within this segment remains \(0 \leq \sin x \leq 1\). The sine function creates a smooth transition between the values within its range, except where the greatest integer function introduces jumps. These jumps occur at points where \([n+p\sin x]\) suddenly moves to the next integer value. Understanding these jumps is critical when studying the differentiability of such functions, as it highlights the points where calculus "breaks" if approached traditionally.