Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f\left(\frac{x+y}{2}\right)=\frac{1}{2}[f(x)+f(y)]\) for real \(x\) and \(y .\) If \(f^{\prime}(0)\) exists and equals \(-1\) and \(f(0)=1\) then the value of \(f(2)\) is(A) 1 (B) \(-1\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The value of \(f(2)\) is -1.

Step 1: Understand the Functional Equation

We are given the functional equation \(f\left(\frac{x+y}{2}\right)=\frac{1}{2}[f(x)+f(y)]\). This is the definition of a Jensen function, which implies that \(f(x)\) is likely linear if it has a derivative everywhere.

Step 2: Explore Properties of Linear Functions

A linear function takes the form \(f(x) = ax + b\). We are given \(f^{\prime}(0) = -1\), which hints at a slope \(a = -1\). Also, \(f(0) = 1\), indicating \(b = 1\). Thus, \(f(x) = -x + 1\).

Step 3: Derive \(f(x)\) Using Given Conditions

Plug the form of the linear function into the given condition: since \(f(x) = -x + 1\), we confirm: \(f^{\prime}(x) = a = -1\), and \(f(0) = 1\). Both conditions have been met.

Step 4: Calculate \(f(2)\)

Using the derived function \(f(x) = -x + 1\), calculate \(f(2)\):\[ f(2) = -2 + 1 = -1 \]. So, the value of \(f(2)\) is \(-1\).

Key Concepts

Linear Functions

Linear functions are fundamental in mathematics. They are straightforward functions that graph as a straight line. Generally, a linear function can be expressed in the form \( f(x) = ax + b \), where \( a \) is the slope and \( b \) is the y-intercept.

• The slope \( a \) indicates how steep the line is. A positive slope means the line ascends as \( x \) increases, while a negative slope indicates the line descends.
• The y-intercept \( b \) is the point where the line crosses the y-axis. This occurs when \( x = 0 \), making \( f(0) = b \).

In the context of Jensen's Functional Equation, recognizing the function as linear can be helpful. When we determine \( f(x) = -x + 1 \), we see that it satisfies the equation by being linear, with the slope \( a = -1 \) and intercept \( b = 1 \). This captures the relationship described in the problem.

Derivative

The derivative of a function expresses how the function's value changes with respect to changes in the input \( x \). It is a crucial concept in calculus. In mathematical terms, the derivative of \( f(x) \) at a point \( x \) is given by the limit:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

• The derivative can be understood as the slope of the tangent line to the curve at any given point.
• When the derivative \( f'(0) = -1 \), it means that near \( x = 0 \), the function is decreasing at a rate of \(-1\) per unit increase in \( x \).

In our examination of the problem, knowing that \( f'(0) = -1 \) helps us to deduce the linearity of the function, verifying that \( f(x) = -x + 1 \) is consistent with this derivative property.

Functional Analysis

Functional analysis is a branch of mathematics dealing with functions and their properties. It extends concepts from algebra and calculus to more complex systems of functions and is vital in understanding equations like Jensen's Functional Equation.
Considering Jensen's Functional Equation, \( f\left(\frac{x+y}{2}\right) = \frac{1}{2}[f(x) + f(y)] \), we note how it describes a balance condition typical in analysis. Such an equation often reveals symmetrical properties:

• The equation suggests that \( f \) averages its output over the inputs \( x \) and \( y \).
• For a linear function, this shows consistency across transformations, a particular interest in functional analysis.

This particular functional equation implies linearity when \( f \) has a derivative, leading us to derive \( f(x) = -x + 1 \) effectively in our problem. Understanding the principles of functional analysis helps to appreciate why these properties hold in more intricate mathematical structures.