Problem 53 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function such that \(|f(x)| \leq x^{2}\), for all \(x \in R .\) Then, at \(x=0\), fis: \(\quad\) (a) continuous but not differentiable. (b) continuous as well as differentiable. (c) neither continuous nor differentiable. (d) differentiable but not continuous.

Step-by-Step Solution

Answer

The function is both continuous and differentiable at \(x=0\). (Option b)

Step 1: Analyze Continuity at x=0

To determine continuity at \(x=0\), we need to verify if the function \(f(x)\) is continuous there. A function is continuous at \(x=0\) if \(\lim_{{x \to 0}} f(x) = f(0)\). Given \(|f(x)| \leq x^2\), all values of \(f(x)\) are squeezed between \(-x^2\) and \(x^2\), both of which approach \(0\) as \(x\) approaches \(0\). Hence, by the Squeeze Theorem, \(\lim_{{x \to 0}} f(x) = 0\) and \(f(0) = 0\), so \(f(x)\) is continuous at \(x=0\).

Step 2: Check Differentiability at x=0

To check differentiability at \(x=0\), we calculate the derivative \(f'(0)\) using the definition: \(f'(0) = \lim_{{h \to 0}} \frac{f(h) - f(0)}{h}\). Since \(f(0) = 0\), \(f'(0) = \lim_{{h \to 0}} \frac{f(h)}{h}\). Since \(|f(h)| \leq h^2\), we can write \(-h^2 \leq f(h) \leq h^2\). Dividing by \(h\) gives \(-h \leq \frac{f(h)}{h} \leq h\). As \(h\) approaches \(0\), both \(-h\) and \(h\) approach \(0\). Hence, by the Squeeze Theorem, \(\lim_{{h \to 0}} \frac{f(h)}{h} = 0\). Therefore, \(f(x)\) is differentiable at \(x=0\) with \(f'(0) = 0\).

Step 3: Conclude the Nature of f at x=0

Since \(f(x)\) is both continuous and differentiable at \(x=0\), the nature of \(f\) at this point is confirmed. There isn’t any contradiction with the definitions of continuity and differentiability. Hence, the correct option is (b) continuous as well as differentiable.

Key Concepts

Function Analysis

Function analysis is a technique used to study the behavior and properties of functions. In the given exercise, we look at the function \(f(x)\) to understand its nature at \(x = 0\). Analyzing a function involves evaluating limits, continuity, and differentiability, among other characteristics.

As part of function analysis, we look at how a function behaves close to a point of interest, in this case, \(x=0\). By examining the inequality \(|f(x)| \leq x^2\), we see that the function's values are bound within a range - they cannot exceed the square of x in magnitude. This constraint immediately guides us as we explore properties like continuity and differentiability.

Function analysis is essential because it provides a foundation for solving complex equations and understanding mathematical phenomena. It helps break down functions into simpler forms, allowing us to apply theorems and calculus strategies effectively.

Squeeze Theorem

The Squeeze Theorem is a fundamental tool in calculus used to find limits. It is particularly useful when a function is trapped between two other functions that converge to the same limit. In this problem, we have \(|f(x)| \leq x^2\), which means that \(-x^2 \leq f(x) \leq x^2\).

As \(x\) approaches \(0\), both \(-x^2\) and \(x^2\) approach \(0\). The Squeeze Theorem helps us conclude that \( ext{lim}_{x \to 0} f(x) = 0\) because \(f(x)\) is 'squeezed' to \(0\) in the limit.

The application of this theorem is not only a mathematical exercise but also demonstrates a deeper analysis of function behavior. This theorem ensures a logical transition from the behavior of bounding functions to inferences about the function of interest.

Limits and Continuity

Understanding limits and continuity is crucial in analyzing functions. A limit involves the value that a function's output approaches as its input nears a certain point. For a function \(f(x)\) to be continuous at a point \(x = 0\), the limit as x approaches 0 must equal the function's value at that point: \( ext{lim}_{x \to 0} f(x) = f(0)\).

In our example, given \(|f(x)| \leq x^2\), both the lower bound \(-x^2\) and upper bound \(x^2\) approach 0 as x does. Hence, by the Squeeze Theorem, \( ext{lim}_{x \to 0} f(x) = 0\) and because it aligns with \(f(0)\) which is also \(0\), \(f(x)\) is continuous at \(x=0\).

Understanding how limits support the concept of continuity helps students appreciate why a function can behave predictably around specific points. It reassures us about function stability when moving infinitesimally close to a point.

Differentiability at a Point

Differentiability at a point is about whether we can find a derivative for the function at that specific location. A function is considered differentiable at \(x=0\) if the derivative exists at that point. The derivative, \(f'(0)\), is determined using the formula: \(f'(0) = ext{lim}_{h \to 0} \frac{f(h) - f(0)}{h}\).

When \(f(0) = 0\), the problem simplifies to finding the limit \( ext{lim}_{h \to 0} \frac{f(h)}{h}\). Due to the constraint \(-h^2 \leq f(h) \leq h^2\), dividing each part by \(h\), we get \(-h \leq \frac{f(h)}{h} \leq h\). As \(h \to 0\), we see that \(-h\) and \(h\) both approach \(0\). Applying the Squeeze Theorem again ensures the limit is \(0\), meaning \(f'(0) = 0\).

This means \(f(x)\) is differentiable at \(x=0\), as the limit exists and is finite. Recognizing differentiability supports understanding function changes and rates of change at exact points, strengthening calculus foundations.