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

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) \(f'(x) = a\).

Step 1: Understand the functional equation

The exercise presents a functional equation where after substituting an arithmetic mean into the function \(f\), it equals the arithmetic mean of the function values. The general form of the equation is \(f\left(\frac{x_1 + x_2 + \ldots + x_n}{n}\right) = \frac{f(x_1) + f(x_2) + \ldots + f(x_n)}{n}\).

Step 2: Consider a specific case

Consider the special case where \(n=1\). In this case, the equation simplifies to \(f(x_1) = f(x_1)\), which holds trivially for any \(f\). To gain more insight, it's useful to consider \(n=2\) and perform further analysis.

Step 3: Simplify for two variables

For \(n=2\), the equation becomes \(f\left(\frac{x_1 + x_2}{2}\right) = \frac{f(x_1) + f(x_2)}{2}\). This is a well-known property of linear functions, suggesting that \(f\) is a linear function.

Step 4: Determine the form of f(x)

Since the equation suggests linearity, assume \(f(x) = mx + c\), where \(m\) and \(c\) are constants. This assumption satisfies the given functional equation, because both sides of the equation will simplify to \(\frac{mx_1 + mx_2 + 2c}{2}\).

Step 5: Differentiate and use given conditions

Differentiate \(f(x) = mx + c\) to find \(f'(x) = m\). Use the given conditions \(f'(0) = a\) and \(f(0) = b\). From \(f(x) = mx + c\), we have \(f'(x) = m = a\) and \(c = b\).

Step 6: Conclusion

Since \(f'(x) = a\) everywhere for the linear form \(f(x) = ax + b\), we conclude that \(f'(x)\) is equal to \(a\).

Key Concepts

Arithmetic Mean

The arithmetic mean is one of the most common ways to find the average of a set of numbers. Simply put, it's the sum of the numbers divided by the count of numbers. For instance, the arithmetic mean of the numbers 2, 4, and 6 would be calculated as \( \frac{2 + 4 + 6}{3} = 4 \).
You encounter the arithmetic mean often in statistics, and it's a straightforward method to summarize a set of data with a single number.
In functional equations, applying the arithmetic mean can help us simplify equations or identify properties, especially when evaluating functions at specific points. By substituting the arithmetic mean into functions, we can analyze the behavior of the function across different inputs. This is useful for determining whether a function is linear or exhibits other properties.

Differentiable Functions

Differentiable functions are a cornerstone of calculus and concern functions that have a derivative at every point within their domain. The derivative represents the rate of change of the function with respect to its variable.
If a function is differentiable at a point, it must be continuous there, but not all continuous functions are necessarily differentiable. For instance, the function \(f(x) = |x|\) is continuous everywhere but not differentiable at \(x = 0\).
In the context of our original exercise, knowing that the function \(f(x)\) is differentiable means we can compute its derivative, \(f'(x)\).
This derivative is crucial in the final step where we identify the constants of the linear function in relation to the given conditions like \(f'(0)=a\) and \(f(0)=b\). Understanding differentiability helps us grasp how \(f(x)\) behaves and changes.

Linear Functions

Linear functions are among the simplest forms of functions. They can be written in the form \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This straightforward form makes them easy to analyze and solve.
Linear functions graph as straight lines, which means their rate of change is constant. This characteristic matches perfectly with the functional equation given in the problem, which suggests that \(f\) might be linear since the equation maintains the structure when the arithmetic mean is applied.
The differentiation of a linear function, \(f'(x) = m\), tells us that the derivative is constant. Therefore, finding that \(f'(x)=a\) in the context of our exercise confirms that the function is linear and that its slope is equal to \(a\).
Knowing the properties of linear functions, along with applying conditions like \(f(0) = b\), helps us determine the characteristics of \(f(x)\) effectively.