Problem 49 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{S}=\left\\{\mathrm{t} \in \mathrm{R}: \mathrm{f}(\mathrm{x})=|\mathrm{x}-\pi|\left(\mathrm{e}^{|\mathrm{x}|}-1\right) \sin |\mathrm{x}|\right.\) is not differentiable at \(\mathrm{t}\\}\). Then the set \(\mathrm{S}\) is equal to: (a) \(\\{0\\}\) (b) \(\\{\pi\\}\) (c) \(\\{0, \pi\\}\) (d) \(\phi\) (an empty set)

Step-by-Step Solution

Answer

The set \(S = \{0, \pi\}\), so the answer is (c).

Step 1: Understanding Differentiability

A function is not differentiable at a point if it is either discontinuous or if its derivative does not exist at that point. In this exercise, we need to find such points for the function \( f(x) = |x-\pi| (e^{|x|} - 1) \sin |x| \).

Step 2: Checking Differentiability at Key Points

Common points to check for differentiability in absolute value functions are where the expression inside the modulus is zero, as this is where the piecewise definition changes. For \( |x-\pi| \), x = \(\pi\). Also, since the other part has \( |x| \), check at x = 0.

Step 3: Examining x = 0 for Differentiability

Observe the function around x = 0. The function \( f(x) \) involves \(|x|\) and \(\sin |x|\), both of which introduce non-differentiability at x = 0 due to a cusp formed by \(|x|\) and non-smoothness of \(\sin |x|\). Therefore, \(f(x)\) is not differentiable at \(x = 0\).

Step 4: Examining x = \pi for Differentiability

At x = \pi, the function involves \(|x - \pi|\), which changes definition at x = \pi. The function will likely have a sharp point (cusp) there, making it non-differentiable at \(x = \pi\).

Step 5: Conclusion on Points of Non-Differentiability

Having found points 0 and \(\pi\) where \(f(x)\) is not differentiable, the function \(f(x)\) is non-differentiable at both these points. Thus, the set S includes both 0 and \(\pi\).

Key Concepts

Absolute Value Function

The concept of the absolute value function is significant in understanding differentiability, especially because it can lead to non-differentiable points. The absolute value of a number or expression, denoted by vertical bars such as \(|x|\), represents its distance from zero on the real number line, always yielding a non-negative result.
For example, \(|3| = 3\) and \(|-3| = 3\).
The absolute value function is a piecewise function, defined as:

• \(|x| = x\) when \(x \geq 0\)
• \(|x| = -x\) when \(x < 0\)

This definition results in a V-shaped graph, where the vertex at the origin creates a sharp turn, or cusp, which indicates where it can become non-differentiable.
This is because the slope of the function changes abruptly, which prevents the existence of a tangent line at this point.

Derivative

Understanding the derivative is crucial in determining where a function is differentiable. The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
It's like finding the slope of the tangent line that just touches the graph at that specific location. Mathematically, the derivative is defined by the limit:\[\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]The function described in the exercise \(f(x) = |x-\pi|(e^{|x|} - 1)\sin|x|\) involves derivatives of absolute value and trigonometric expressions.
Calculating derivatives for such functions can be complex due to their piecewise nature and the potential for non-smooth transitions, such as cusps caused by the absolute value.

Piecewise Function

Piecewise functions are another essential concept in understanding differentiability. A piecewise function is defined by different expressions for different intervals of the domain.
For example, the absolute value function is piecewise because it applies a different rule depending on whether the input is positive or negative.
In terms of differentiability, the points where these different expressions meet, such as the edges of the intervals, are critical.
At these points, it is necessary to check for continuity and whether the derivatives from either side agree. If not, the function may not be differentiable at these junctions, leading to sharp turns or discontinuities.

Non-Differentiable Point

A non-differentiable point is where a function does not have a derivative. This can happen for several reasons.
One reason is discontinuity—if a graph jumps or has a hole, it can't possess a tangent line there.
Additionally, a function with a cusp (sharp point) or vertical tangent cannot have a well-defined slope, hence no derivative.
In the exercise, the non-differentiable points were identified at 0 and \(\pi\), due to the structure of the absolute value function.
This function changes its approach at these points, leading to cusps and thus non-differentiability. Recognizing these points is key in analyzing the behavior of functions involving absolute values.