Problem 76 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A function \(f: R \rightarrow R\), where \(R\) is the set of real numbers satisfies the equation \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)+f(0)}{3}\) for all \(x, y\) in \(R\). If the function \(f\) is differentiable at \(x=\) 0 , then \(f\) is (A) linear (B) quadratic (C) cubic (D) biquadratic

Step-by-Step Solution

Answer

A linear function satisfies the given conditions.

Step 1: Substitute to Simplify

Let x = y = 0 in the given equation. We have \( f\left(\frac{0+0}{3}\right) = \frac{f(0) + f(0) + f(0)}{3} \). Simplifying gives \( f(0) = f(0) \), which is trivially true.

Step 2: Consider Derivative and Limit Properties

Assume \( f(x) = ax + b \), which is a linear function and thus differentiable at any point, including zero. Notice if \( f(x) \) were a higher degree polynomial, it would not maintain the form of the equation when differentiated because differentiability at zero with the consistent equation behavior suggests a linear form without remaining degree terms.

Step 3: Check Functional Equation for Linear Solution

Assume a linear solution \( f(x) = ax + b \). Substitute into the functional equation: \[ a\left(\frac{x+y}{3}\right) + b = \frac{(ax + b) + (ay + b) + b}{3} \]. Simplify both sides to confirm: \[ \frac{ax + ay + 3b}{3} = \frac{ax + ay + 3b}{3} \], which holds true.

Step 4: Conclusion of Function's Form

The substitution confirms that a linear function like \( f(x) = ax + b \) satisfies the given condition for all \( x, y \) in \( R \). Hence, the form of the function is necessarily linear to fulfill the derivational consistency and equation.

Key Concepts

Differentiable Functions

Differentiable functions are those functions that have a derivative at every point in their domain. In simpler terms, a function is differentiable if you can draw a tangent to its curve at any point without lifting your pencil. The derivative of a function at a given point tells us how the function changes; specifically, it gives us the slope of the tangent line at that point.

Some important points about differentiable functions include:

• Differentiability implies continuity. If a function is differentiable at a point, it is also continuous at that point. However, being continuous does not necessarily mean it is differentiable.
• For a function to be differentiable, its derivative must exist, and this involves calculating the limit of the difference quotient as the interval approaches zero.
• In calculus, common differentiable functions include linear functions, polynomial functions, exponential functions, and trigonometric functions.

Understanding differentiable functions involves observing how slopes and curves behave; if the slope of a curve changes abruptly or creates a corner or cusp, the function may not be differentiable at that point.

In the context of this exercise, the function's differentiability at zero ensures the function behaves in a stable and predictable manner around this point, guiding us towards understanding its form.

Linear Functions

Linear functions are algebraic expressions of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. These functions create straight-line graphs and are characterized by a constant rate of change or slope, represented by \( a \). Linear functions are foundational in mathematics due to their simplicity and ease of use in modeling relationships between variables.

Here are some key characteristics of linear functions:

• They are the simplest form of polynomial functions.
• The graph of a linear function is a straight line, hence the name 'linear.'
• The slope, \( a \), tells us how steep the line is and the direction it goes. If \( a \) is positive, the line slopes upwards; if negative, it slopes downwards.
• The y-intercept, \( b \), is the point where the line crosses the y-axis when \( x = 0 \).

In the given exercise, recognizing that \( f(x) = ax + b \), a linear function, satisfies the functional equation highlights its role in maintaining consistency when differentiable at zero. Unlike higher-degree polynomials, linear functions do not accumulate additional terms upon differentiation, making them suitable for satisfying the given equation.

Real Numbers

Real numbers are a vast set of numbers typically encountered in everyday mathematics. They include all the numbers that can be found on a number line, ranging from negative numbers to positive ones, including whole numbers, fractions, and irrational numbers like \( \pi \) and \( e \).

Let’s explore some essential aspects of real numbers:

• The set of real numbers is denoted by \( R \).
• Real numbers are foundational to calculus and continuous mathematics, as they fill the gaps on the number line without any breaks.
• Subsets of real numbers include rational numbers (fractions like 1/2, 3/4), and irrational numbers (like \( \sqrt{2} \) and \( \pi \)).
• All real numbers are represented as infinite decimal expansions; for example, 2.5, -3.142, and 7.

In mathematical problems, like the one in our original exercise, real numbers provide a broad domain within which functions like \( f: R \rightarrow R \) can operate. This allows for extensive mapping of inputs to outputs, highlighting the function's behavior across an unbroken numerical spectrum. This ensures that functional equations involving real numbers are applicable in a wide range of scenarios.