Problem 54 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The function is not differentiable at \(-1, 0, 1, 2, 3\). Option (A) is correct.

Step 1: Understand the Components

The function is given as \( f(x) = [x] + |1-x| \), where \([-1 \leq x \leq 3]\) and \([x]\) denotes the greatest integer less than or equal to \(x\). The greatest integer function, also known as the floor function, can cause discontinuities at integer values of \(x\). The absolute value function changes its behavior at the point where its argument is zero, in this case at \(x=1\).

Step 2: Analyze the Floor Function

The greatest integer function \([x]\) is not continuous at integer values of \(x\) because it jumps between integers. Therefore, \(f(x)\) would be non-differentiable at integer values within the given interval \([-1, 3]\). These integers are \(-1, 0, 1, 2, 3\).

Step 3: Analyze the Absolute Value Function

The absolute value function \(|1-x|\) is not differentiable at the point where the argument is zero, i.e., where \(1-x=0\). This occurs at \(x=1\), which is inside our interval.

Step 4: Combine Points of Non-Differentiability

Combine the points obtained from the analysis of both \([x]\) and \(|1-x|\). Considering both, the non-differentiable points within \([-1, 3]\) due to \([x]\) are \(-1, 0, 1, 2, 3\) and due to absolute value at \(x=1\). Thus, the combined set remains \(-1, 0, 1, 2, 3\).

Step 5: Conclusion

Based on the analysis, the function is not differentiable at the points \(-1, 0, 1, 2, 3\). Therefore, the correct answer is option (A).

Key Concepts

Greatest Integer Function

When you see a notation like \([x]\), this indicates the greatest integer function, also known as the floor function. This function essentially "rounds down" any real number to the largest integer below it. For example:

• For \(x = 2.3\), \([x] = 2\)
• For \(x = -1.2\), \([x] = -2\)

The significant character of this function is that it remains constant over intervals and abruptly jumps at integer points. This jump is what causes the function to be discontinuous at these points. For instance:

• At \(x = 1\), it jumps from 0 to 1.
• At \(x = 2\), it jumps from 1 to 2.

These jumps imply that the derivative doesn't exist at integers, making such points non-differentiable. In our interval \([-1, 3]\), the greatest integer function affects differentiability at \(-1, 0, 1, 2,\) and \(3\).

Absolute Value Function

The absolute value function, represented by \(|x|\), is fundamental in expressing the "magnitude" or "distance" of a number from zero, without regard to direction or sign. This means:

• For positive input, \(|x| = x\)
• For negative input, \(|x| = -x\)

Within our problem, we have \(|1-x|\), which changes its behavior based on whether \((1-x)\) is positive or negative. The "switch" occurs precisely when \(1-x=0\) or \(x=1\). At this exact point, the function changes from being linear with a positive slope to linear with a negative slope.
This change in slope at \(x=1\) means that the function is not smooth and hence, not differentiable at this point. Beyond this point, the function continues to be linear on either side, but the derivative (slope) changes direction, creating a juncture of non-differentiability.

Points of Non-Differentiability

In calculus, differentiability refers to whether a function has a derivative, which implies the function is smooth. When a function lacks a derivative at a point, it is said to be non-differentiable at that point. Common causes of non-differentiability include jumps, corners, cusps, or vertical tangents.
For the function in question, \(f(x) = [x] + |1-x|\), both components contribute to points of non-differentiability:

• The greatest integer function \([x]\) leads to non-differentiability at all integers within the interval, specifically at \(-1, 0, 1, 2,\) and \(3\).
• The absolute value function \(|1-x|\) adds a point of non-differentiability at \(x=1\), where it changes its form.

By combining these insights, we conclude that the complete set of non-differentiability points for \([-1, 3]\) is \(-1, 0, 1, 2, 3\). Thus, understanding and identifying these points helps in grasping where and why the function loses differentiability.