Problem 35 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f\left(\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right)=\) \(\frac{f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots+f\left(x_{n}\right)}{n}\) where all \(x_{i} \in R\) are independent to each other and \(n \in N\). If \(f(x)\) is differentiable and \(f^{\prime}(0)=a, f(0)=b\) then \(f^{\prime}(x)\) is equal to (A) \(a\) (B) 0 (C) \(b\) (D) None of these

Step-by-Step Solution

Answer

(A) a

Step 1: Understand the Functional Equation

We are given a functional equation \( f\left(\frac{x_1 + x_2 + \ldots + x_n}{n}\right) = \frac{f(x_1) + f(x_2) + \ldots + f(x_n)}{n} \). The function must satisfy this condition for all independent real numbers \( x_i \text{ and } n \in \mathbb{N} \). This implies that \( f(x) \) is linear.

Step 2: Analyze the Properties of the Function

Since \( f(x) \) is differentiable and given that \( f'(0) = a \text{ and } f(0) = b \), these suggest \( f(x) \) has the form \( f(x) = ax + b \). This is because a linear function is the simplest form that can be both differentiable everywhere and satisfy the functional equation across all combinations of inputs.

Step 3: Verify the Form of \( f(x) \)

To ensure that \( f(x) = ax + b \) satisfies the given functional equation, substitute it into both sides of the equation. The left side becomes \( f\left(\frac{x_1+x_2+\ldots+x_n}{n}\right) = a\frac{x_1+x_2+\ldots+x_n}{n} + b \). The right side becomes \( \frac{f(x_1) + f(x_2) + \ldots + f(x_n)}{n} = \frac{(ax_1+b) + (ax_2+b) + \ldots + (ax_n+b)}{n} \), which simplifies to the same expression. Thus, the form holds.

Step 4: Differentiate to Find \( f'(x) \)

Differentiate \( f(x) = ax + b \) with respect to \( x \). The derivative \( f'(x) = \frac{d}{dx}(ax + b) = a \). This confirms that the derivative is constant and equal to \( a \).

Step 5: Match Solution with Options

From the options provided, (A) \( a \) matches our calculated derivative \( f'(x) = a \). Hence, \( f'(x) \) is equal to \( a \).

Key Concepts

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. This means it is smooth and has no sharp edges or sudden changes in direction. A function being differentiable implies continuity but not vice versa; a continuous function might not have a derivative at every point. The concept of differentiability is crucial in calculus as it allows us to understand how a function changes at any given point. For the function in our problem, the fact that it's differentiable means we can find the derivative, giving us insights into how the function behaves around any point, specifically at zero in this case.

• A function is differentiable if the limit of the difference quotient exists as the interval considered approaches zero.
• A key property of differentiable functions is that they must also be continuous.
• For linear functions, the derivative is constant, which simplifies analysis.

Linear Function

Linear functions are functions of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. These functions graph as straight lines. The key characteristic of linear functions is their constant rate of change, represented by the slope \( a \).In the context of functional equations, linearity makes calculation and verification more straightforward. Linearity ensures that if you take any two points on the line, the rate at which the function changes between these points remains the same.

• The slope \( a \) indicates the steepness of the line.
• The y-intercept \( b \) represents the point where the line crosses the y-axis.
• Linear functions are simple to differentiate, as their derivative remains constant, which is what we see in the solution.

Calculus

Calculus is a branch of mathematics focusing on change and motion. It provides tools for understanding how things move and grow. The two main operations in calculus are differentiation and integration.Differentiation allows us to find the rate at which a quantity changes, which is why it was used in the solution to find \( f'(x) \). Understanding calculus is crucial since it aids in modeling real-world phenomena beyond mere numbers and shapes, extending into dynamic systems like velocity and acceleration.

• Differentiation is used to find the derivative, or the instantaneous rate of change, of functions.
• Integration, the reverse process of differentiation, sums quantities and can determine areas under curves.
• Calculus underpins many advanced topics and applications in physics, engineering, and economics.