Problem 110 - 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 \(f(x+y)=f(x) f(y)\) for all \(x, y\), where \(f(0) \neq 0 .\) If \(f^{\prime}(0)=2\), then \(f(x)=A e^{2 x}\), where \(A\) is a constant. Reason: \(f^{\prime}(x)=f(x)\)

Step-by-Step Solution

Answer

C: Assertion is True, Reason is False.

Step 1: Analyze the Assertion

The assertion states that if a function \( f(x+y) = f(x)f(y) \) for all \( x, y \) with \( f(0) eq 0 \) and \( f'(0) = 2 \), then \( f(x) = A e^{2x} \) where \( A \) is a constant. This suggests that \( f(x) \) could be an exponential function based on the property \( f(x+y) = f(x)f(y) \), which often indicates an exponential form. We need to verify if \( f'(0) = 2 \) confirms \( f(x) = Ae^{2x} \).

Step 2: Differentiate the Exponential Form

Consider \( f(x) = e^{kx} \). Differentiating it gives \( f'(x) = ke^{kx} \). At \( x = 0 \), \( f'(0) = ke^0 = k \). We know \( f'(0) = 2 \), so \( k = 2 \). Thus, the function is \( f(x) = e^{2x} \). Since \( f(x+y) = f(x)f(y) \) holds for \( e^{2x} \), our assertion is consistent if \( f(x) = Ae^{2x} \), with \( A \) being the initial value \( f(0) \).

Step 3: Evaluate the Reason

The reason given is \( f'(x) = f(x) \). For exponential functions like \( f(x) = Ae^{kx} \), \( f'(x) = Ake^{kx} \). Clearly, \( f'(x) = ke^{kx} \), not \( f(x) \), unless \( k = 1 \). Since \( k = 2 \) here, \( f'(x) eq f(x) \), which renders the reason false for \( k = 2 \).

Step 4: Conclusion Based on Assertion and Reason

The Assertion is true because given \( f'(0) = 2 \), the function can be \( f(x) = Ae^{2x} \). However, the Reason is false because \( f'(x) = f(x) \) is not correct as the derivative is \( 2Ae^{2x} \), not the same as the original function.

Key Concepts

Assertion and Reasoning Questions

Understanding **Assertion and Reasoning Questions** involves evaluating whether both the assertion and the reason are true or false, and if true, checking if the reason correctly explains the assertion. These questions are designed to test your comprehension of concepts and the ability to correlate them logically.

• Begin by examining the **assertion** independently. Check if the statement is true based on the information given or your knowledge.
• Next, assess the **reason**. Again, determine its truthfulness separately.
• The crucial part is determining if the reason provides a valid explanation for the assertion. Is it merely true alongside it, or does it actively justify why the assertion is correct?

Employing this approach effectively can refine your problem-solving skills, as it encourages detailed analysis and reasoning. This type of question often appears in exams like the JEE Mathematics due to its ability to distinguish between superficial understanding and deep conceptual knowledge.

Exponential Functions

Exponential functions are a key concept in mathematics, particularly in calculus and algebra. They are of the form \( f(x) = Ae^{kx} \), where \( A \) and \( k \) are constants. These functions exhibit growth or decay depending on the value of \( k \). For example:

• If \( k > 0 \), the function shows **exponential growth**.
• If \( k < 0 \), the function represents **exponential decay**.

Let's delve deeper:
- **Unique Property**: Exponential functions have the special property \( f(x+y) = f(x)f(y) \), which aligns with the assertion given in the problem. This property holds true because of the behavior of exponential calculations.- **Initial Conditions**: In many problems, the initial condition \( f(0) eq 0 \) ensures that the constant \( A \) remains meaningful and helps in determining the entire function based on the given derivatives or other conditions.
Exponential functions are widely used in modeling real-world phenomena such as population growth, radioactive decay, and financial interest calculations because they naturally describe processes that change exponentially over time.

Differential Calculus

**Differential calculus** deals with the concept of the derivative, which represents the rate at which a function is changing at any given point. It is a fundamental tool for understanding the behavior of functions.

• The **derivative** of a function \( f(x) \), denoted \( f'(x) \), can be understood as the slope of the tangent line to the curve at a point.
• For an exponential function like \( e^{kx} \), its derivative is \( k e^{kx} \). Hence, in the problem, \( f'(x) = 2e^{2x} \) helps verify the assertion.

Here's a deeper understanding:
- **Interpretation of Derivatives**: Derivatives are essential for examining how functions behave. For instance, a constant derivative indicates linear behavior, while a variable derivative suggests more complex, often curved, behavior.
- **Applications**: Differential calculus is vital for optimization problems, physics motion equations, and growth models, where determining maximum and minimum values or rate of change is crucial.- **Conclusion from the Problem**: Understanding that \( f'(0) = 2 \) leads us to find \( f(x) = Ae^{2x} \) reflects the essential nature of how derivatives guide us in determining the type of function we are dealing with.Differential calculus is a core tool in JEE Mathematics and is invaluable in solving real-life problems that involve change.