Problem 111 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\max .\left\\{x, x^{3}\right\\} .\) Then, \(f(x)\) is not differentiable at \(x=-1,0,1\) Reason: \(f(x)=\left\\{\begin{array}{l}x, x \leq-1 \\ x^{3},-11\end{array}\right.\)

Step-by-Step Solution

Answer

A, both the assertion and reason are correct, and the reason correctly explains the assertion.

Step 1: Understanding Assertion (A)

The assertion states that the function \(f(x) = \max \{x, x^3\}\) is not differentiable at \(x = -1, 0, 1\). To explore this, we evaluate \(f(x)\) around these points. \(f(x)\) is determined by the maximum of \(x\) and \(x^3\). The function switches from one form to another at these points, suggesting that they are candidates for non-differentiability.

Step 2: Analyzing the Behavior at x = -1

At \(x = -1\), \(f(x) = x = -1\) for \(x \leq -1\) and \(f(x) = x^3\) for \(-1 < x \leq 0\). The function switches from \(x\) to \(x^3\) at \(x = -1\), creating a kink, which means the function is not differentiable at \(x = -1\).

Step 3: Analyzing the Behavior at x = 0

At \(x = 0\), \(f(x) = x^3 = 0\) for \(-1 < x \leq 0\) and \(x = 0\) for \(0 < x \leq 1\). The function switches from \(x^3\) to \(x\) at \(x = 0\), also indicating a non-differentiable point.

Step 4: Analyzing the Behavior at x = 1

At \(x = 1\), \(f(x) = x = 1\) for \(0 < x \leq 1\) and \(f(x) = x^3\) for \(x > 1\). Again, \(f(x)\) changes from \(x\) to \(x^3\) at \(x = 1\), making it non-differentiable here as well.

Step 5: Evaluating Reason (R)

The reason provides a piecewise definition: \(f(x)\) is \(x\) for \(x \leq -1\), \(x^3\) for \(-1 < x \leq 0\), \(x\) for \(0 < x \leq 1\), and \(x^3\) for \(x > 1\). This accurately describes the function and its behavior at \(x = -1, 0, 1\), supporting the assertion. However, the explanation that these points are where \(x\) and \(x^3\) switch does not specifically assert non-differentiability, only conditions for it.

Key Concepts

Piecewise Functions

Piecewise functions are a fascinating mathematical concept where a function is defined by different expressions for different parts of its domain. They allow us to describe functions that have varying behaviors across different intervals. For example, the piecewise function defined in the exercise,

• uses the expression \( x \) for some parts of the domain (such as \( x \leq -1 \) and \( 0 < x \leq 1 \)),
• and \( x^3 \) for others (like \( -1 < x \leq 0 \) and \( x > 1 \)).

Piecewise functions are useful in modeling situations where a single expression cannot adequately describe a process over all values of \( x \). For example, they can model real-world phenomena like traffic patterns, stock prices, or growth rates, which may change dramatically over different conditions. In mathematics, analyzing such functions requires paying close attention to the intervals over which each expression holds true.

Function Analysis

Function analysis is the study of the various properties of functions, such as their continuity, differentiability, and overall behavior. To analyze a function, one often examines:

• The definition of the function and its domain,
• Where the function is increasing or decreasing,
• Points of continuity and discontinuity,
• And importantly, where the function is differentiable.

In our exercise, we delve into the function \(f(x) = \max \{x, x^3\}\), uncovering its characteristics by comparing its different pieces. This function is intriguing because it shifts between two expressions. The shifts between \( x \) and \( x^3 \) at specific points prompt an examination of its differentiability. By examining intervals and transitions, function analysis provides insight into the behavior at critical points like \( x = -1, 0, \) and \( 1 \), where the piecewise segments interchange.

Non-differentiable Points

Non-differentiable points are important aspects of function analysis, particularly when dealing with piecewise functions. A point in a function is non-differentiable if the function has a 'kink' or a sharp corner at that point. Another reason for non-differentiability could be a discontinuity or a vertical tangent.
In the exercise, the assertion that \( f(x) \) is not differentiable at \( x = -1, 0, \) and \( 1 \) is due to the function switching from \( x \) to \( x^3 \). Such switches often result in non-differentiable points.
At \( x = -1 \), \( x = 0 \), and \( x = 1 \), there is a change in the expression used to define \( f(x) \), leading the derivatives from either side to not match at these points. For instance, at \( x = 0 \), the graphs of \( x^3 \) and \( x \) touch without aligning in their slopes, illustrating a classic example of non-differentiability at these transition points.