Problem 54 - A Complete Resource Book in Mathematics for JEE Main

Question

The points where the function \(f(x)=[x]+|1-x|,-1\) \(\leq x \leq 3\), where \([.]\) denotes the greatest integer function, is not differentiable, are (A) \(x=-1,0,1,2,3\) (B) \(x=-1,0,2\) (C) \(x=0,1,2,3\) (D) \(x=-1,0,1,2\)

Step-by-Step Solution

Answer

(A) \(x=-1, 0, 1, 2, 3\).

Step 1: Understanding the Function's Components

The function is given by \( f(x) = [x] + |1-x| \), where \([.]\) denotes the greatest integer (floor) function and \(| \, |\) denotes the absolute value. To solve the problem, we need to identify where the function \( f(x) \) is not differentiable in the given interval \([-1, 3]\).

Step 2: Analyzing the Greatest Integer Function

The greatest integer function \([x]\) is not differentiable at any integer because it has a jump at these points. Within the interval \([-1, 3]\), these points are \(-1, 0, 1, 2, 3\).

Step 3: Analyzing the Absolute Value Term

The expression \(|1-x|\) is not differentiable at \(x=1\) because the derivative of an absolute value function does not exist where the argument is zero (where \(1-x=0\)).

Step 4: Identifying Non-Differentiable Points

For \(f(x) = [x] + |1-x|\), non-differentiable points occur where either component \([x]\) or \(|1-x|\) is not differentiable. We have found that these points are \(-1, 0, 1, 2, 3\).

Step 5: Choosing the Correct Multiple Choice Answer

Checking the options, option (A) states \(x = -1, 0, 1, 2, 3\) which matches all the points found in the analysis. Therefore, option (A) is the correct answer.

Key Concepts

Greatest Integer Function

The greatest integer function, also called the floor function, is denoted by \([x]\). This function outputs the largest integer less than or equal to \(x\). For example, * \([3.7] = 3\) * \([-2.1] = -3\) It's important to note that the greatest integer function is not continuous at integer points, and this is where it becomes non-differentiable. For the greatest integer function, non-differentiability occurs at every integer value because the function has a sudden jump at those places. Imagine trying to draw the function on graph paper: you would have to lift your pencil and move it vertically to the next point to continue. This discrete jump indicates the absence of a tangent, and thus a derivative, at those integer values. Within the interval \([-1, 3]\), it is non-differentiable at the integers \(-1, 0, 1, 2,\) and \(3\).

Absolute Value Function

The absolute value function portrays how far a number is from zero on the number line. Represented as \(|x|\), it returns the non-negative magnitude of \(x\), regardless of its sign. Here's how it behaves: * If \(x\) is positive or zero, \(|x| = x\). * If \(x\) is negative, \(|x| = -x\). In this exercise, we work with \(|1-x|\), which means the expression changes based on whether \(1-x\) is positive or negative. This implies a potential discontinuity in the slope, making the derivative non-existent at the point where the value inside the absolute value equals zero.For \(f(x) = [x] + |1-x|\), the term \(|1-x|\) becomes zero at \(x = 1\). This is where the function's slope changes abruptly, causing non-differentiability at this point.

Non-differentiable Points

Non-differentiable points occur where a function does not have a well-defined tangent line. This happens due to discontinuities, sharp corners, or cusps in the function's graph. In this exercise, the function \(f(x) = [x] + |1-x|\) is examined for non-differentiability over the interval \([-1, 3]\). Here, non-differentiability stems from two separate parts of the function:- The greatest integer function \([x]\) has jumps at integer values, and is non-differentiable at \(-1, 0, 1, 2,\) and \(3\). - The absolute value function \(|1-x|\) presents a sharp point at \(x = 1\), making it non-differentiable there.Thus, the points \(-1, 0, 1, 2,\) and \(3\) are collectively identified as non-differentiable for \(f(x)\). These points indicate changes in the function which prevent the calculation of a tangent line, confirming non-differentiability. The correct answer is hence option (A), covering all these critical points.