Problem 79 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x-y)+f(x+y)=2 f(x) f(y) \forall x, y \in R\), then (A) \(f\) is even (B) \(f\) is odd (C) \(f^{\prime}\) is even (D) \(f^{\prime}\) is odd

Step-by-Step Solution

Answer

(A) or (C).

Step 1: Understanding the Problem

We are given the equation \( f(x-y) + f(x+y) = 2f(x)f(y) \) for all real numbers \( x \) and \( y \). The question asks us to determine whether the function \( f \) or its derivative has even or odd properties.

Step 2: Define Even and Odd Functions

A function \( f(x) \) is even if \( f(-x) = f(x) \), and it is odd if \( f(-x) = -f(x) \). The derivative \( f^{\prime}(x) \) inherits these properties: if \( f \) is even, \( f^{\prime}(x) \) is odd, and if \( f \) is odd, \( f^{\prime}(x) \) is even.

Step 3: Check for Even Function Property

Substitute \( x = 0 \) in the given equation: \( f(0-y) + f(0+y) = 2f(0)f(y) \). This simplifies to \( f(-y) + f(y) = 2f(0)f(y) \). If \( f \) is even, \( f(-y) = f(y) \), implying \( 2f(y) = 2f(0)f(y) \). Assuming \( f(y) eq 0 \), this results in \( f(0) = 1 \).

Key Concepts

Even and Odd Functions

Even and odd functions play a fundamental role in mathematics, especially in problem-solving for exams like the JEE Main. To determine if a function is even, you need to check whether it satisfies the property: \( f(-x) = f(x) \). This means the function is symmetric around the y-axis. By contrast, a function is odd if it satisfies \( f(-x) = -f(x) \), indicating symmetry around the origin.

In our original exercise, substituting \( x = 0 \) transformed the equation to \( f(-y) + f(y) = 2f(0)f(y) \). This is a critical step in checking if the function has even properties. When we test for evenness, if \( f(-y) = f(y) \), it implies that both sides of the equation maintain their symmetry. This condition holds when \( f(0) = 1 \) if \( f(y) eq 0 \).

• This implies symmetry around the y-axis for the function.
• It saves computational resources by reducing the complexity of integral and curve-related problems.

Function Derivatives

Function derivatives are essential for understanding the behavior and rate of change of functions. They are used extensively in JEE Main level mathematics. A derivative, \( f^{\prime}(x) \), of a function \( f(x) \), describes its instantaneous rate of change.

When we check if a function is even or odd, we also need to consider its derivative properties. If the function \( f \) is even, its derivative, \( f^{\prime} \), will be odd. Conversely, if \( f \) is odd, \( f^{\prime} \) will be even.

• Derivatives help identify maximum or minimum points, which are essential in calculus problems.
• The behavior of derivatives is crucial for plotting accurate graphs of functions.
• Understanding derivative symmetry helps solve differential equations more effectively.

JEE Main Mathematics Problems

JEE Main mathematics problems often require proficiency in understanding concepts like even and odd functions and their derivatives. These problems test a student's ability to manipulate and solve high-level mathematical equations systematically.

While tackling such problems, students must:

• Identify whether given functions have properties of evenness or oddness.
• Determine how derivatives of these functions change or retain these properties.
• Leverage this understanding to solve complex problems with accuracy and speed.

For example, in the given exercise, we analyzed whether the function or its derivative is even or odd. Such skills are vital for ensuring success in the JEE Main, as they enhance problem-solving speed and accuracy through symmetry knowledge. Understanding these core concepts aids in breaking down complex questions into manageable parts and solving them logically.