Problem 106 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(f(x)=x|x|, g(x)=\sin x\) and \(h(x)=(g \circ f)(x)\). Then (a) \(\mathrm{h}(\mathrm{x})\) is not differentiable at \(\mathrm{x}=0\). (b) \(\mathrm{h}(\mathrm{x})\) is differentiable at \(\mathrm{x}=0\), but \(\mathrm{h}^{\prime}(\mathrm{x})\) is not continuous at \(\mathrm{x}=0\) (c) \(\mathrm{h}^{\prime}(\mathrm{x})\) is continuous at \(\mathrm{x}=0 \mathrm{but}\) it is not differentiable at \(\mathrm{x}=0\) (d) \(\mathrm{h}^{\prime}(\mathrm{x})\) is differentiable at \(\mathrm{x}=0\)

Step-by-Step Solution

Answer

Option (c) is correct; \(h'(x)\) is continuous at \(x = 0\) and differentiable.

Step 1: Understand Composite Function h(x)

We are given that the function \(h(x)\) is defined as \(h(x) = (g \circ f)(x)\), meaning \(h(x) = g(f(x))\). The function \(f(x) = x|x|\) modifies the input based on the value of \(x\), and \(g(x) = \sin x\) is the sine function.

Step 2: Analyze f(x)

The function \(f(x) = x|x|\) can be expressed as \(f(x) = x^2\) for \(x \geq 0\) and \(f(x) = -x^2\) for \(x < 0\). At \(x = 0\), \(f(0) = 0\). The function is continuous everywhere including at \(x = 0\), but its derivative is not continuous at \(x = 0\).

Step 3: Analyze g(f(x))

Since \(g(x) = \sin x\) is a continuous and differentiable function, when we have \(h(x) = \sin(f(x))\), it remains continuous everywhere because \(\sin(x)\) is continuous for all real numbers.

Step 4: Calculate Derivative of h(x)

Given piecewise input to \(f(x)\), thus \(h(x)\) also becomes piecewise: \(h(x) = \sin(x^2)\) for \(x \geq 0\) and \(h(x) = \sin(-x^2) = \sin(x^2)\) for \(x < 0\). Differentiating, we have: \(h'(x) = 2x \cos(x^2)\) with respect to \(x\), which is continuous at all points including \(x = 0\).

Step 5: Determine Continuity and Differentiability at x = 0

We observe \(h'(x) = 0\) at \(x = 0\), which is continuous at this point. The derivative \(h'(x)\) changes smoothly without jumps or breaks across \(x = 0\). Therefore, \(h(x)\) is differentiable and \(h'(x)\) is continuous at \(x = 0\).

Key Concepts

Composite Function

A composite function is formed by applying one function to the results of another. In our exercise, the function \( h(x) = (g \circ f)(x) \) means placing \( f(x) \) inside \( g(x) \). Thus, \( h(x) = g(f(x)) \). Composite functions are essential when calculating derivatives because you need to understand the output of one function as the input of another.

• For example, if \( f(x) = x|x| \), the function is influenced by the value of \( x \), while \( g(x) = \sin x \) is the sine function. Together, they form \( h(x) \).

Understanding how functions compose helps when analyzing continuity and differentiability because changes in \( f(x) \) directly affect \( g(f(x)) \).

• In this problem, evaluating \( h(x) \) at \( x = 0 \) requires understanding how \( f(x) \) and \( g(x) \) behave around this point.

Continuity

Continuity is the property of a function being smooth and without breaks. A function is continuous at a certain point if the limit of the function, as it approaches that point, equals the function's value at the point.

• At \( x = 0 \), \( f(x) = x|x| \) is continuous because \( f(x) = 0 \) for \( x = 0 \), and it transitions smoothly from positive to negative values as \( x \) crosses zero.

For the sine function \( g(x) = \sin x \), it is continuous everywhere because the sine curve is smooth and oscillates without breaks.

• So, for \( h(x) = g(f(x)) \), since both \( f \) and \( g \) are continuous, \( h(x) \) remains continuous at all points, including \( x = 0 \).

Piecewise Function

A piecewise function is one that has different expressions for different intervals of the input. The function \( f(x) = x|x| \) is an example of a piecewise function, because it can be rewritten as:

• \( f(x) = x^2 \) when \( x \ge 0 \) and
• \( f(x) = -x^2 \) when \( x < 0 \).

This is crucial for the exercise because it defines \( h(x) \) differently based on the interval of \( x \).

• As a piecewise function, \( h(x) = \sin(x^2) \) for all real \( x \), because \( \sin(-x^2) = \sin(x^2) \), making the piecewise definition for \( h(x) \) consistent.

Understanding piecewise functions helps in calculating derivatives and analyzing continuity, as we need to check whether the function could potentially have any breaks or jumps at the boundaries of the pieces.

Derivative

The derivative of a function gives us the rate at which the function is changing at any given point. For \( h(x) = \sin(x^2) \), the derivative \( h'(x) \) indicates how \( h(x) \) changes as \( x \) changes.

• To find the derivative, apply the chain rule: \( h'(x) = 2x \cos(x^2) \).

The derivative's continuity is crucial because if there are no abrupt changes, it indicates that \( h(x) \) is smoothly changing across those points.

• At \( x = 0 \), \( h'(x) = 0 \), showing smooth continuity at this point, with \( h'(x) \) not having any sudden jumps or undefined behaviors.

The problem concludes that \( h(x) \) is differentiable, and its derivative, \( h'(x) \), is continuous at \( x = 0 \). This continuous derivative is a key sign of the function's smoothness as it transitions through \( x = 0 \).