Problem 67 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f\) be a differentiable function satisfying \(f(x+y)=\) \(f(x)+f(y)+x y .\) If \(\lim _{h \rightarrow 0} \frac{1}{h} f(h)=3\), then (A) \(f(x)=3 x\) (B) \(f(x)=3 x+x^{2}\) (C) \(f(x)=3 x+\frac{x^{2}}{2}\) (D) None of these

Step-by-Step Solution

Answer

The function is \( f(x) = 3x + \frac{x^2}{2} \) (Option C).

Step 1: Understanding the Functional Equation

We are given that a function \( f \) satisfies \( f(x+y) = f(x) + f(y) + xy \). This is a generalized form of the Cauchy functional equation with an additional quadratic term \( xy \). Our task is to find \( f(x) \) using this information.

Step 2: Evaluating the Limiting Condition

We are also told that \( \lim_{h \rightarrow 0} \frac{1}{h} f(h) = 3 \). Interpreting this condition, it suggests that the derivative \( f'(0) = 3 \), where this limit implies differentiability and a linear behavior near zero.

Step 3: Testing the Form of the Function

Let's assume \( f(x) \) has the form \( f(x) = ax + bx^2 \). Substituting this into the functional equation gives us \[ f(x+y) = a(x+y) + b(x+y)^2 = ax + ay + b(x^2 + 2xy + y^2) \]while \[ f(x) + f(y) + xy = ax + bx^2 + ay + by^2 + xy \]. Equating both expressions, we find: \[ ax + ay + b(x^2 + 2xy + y^2) = ax + ay + bx^2 + by^2 + xy. \]This implies the coefficient of \( xy \) on both sides must be equal, giving \( 2b = 1 \), thus \( b = \frac{1}{2} \).

Step 4: Determining the Linear Coefficient

Given the condition is \( a = 3 \), derived from \( \lim_{h \to 0} \frac{1}{h}(ah + bh^2) = 3 \), solving gives \( a = 3 \) since the dominant term as \( h \to 0 \) is linear. This forms \( f(x) = 3x + \frac{x^2}{2} \).

Step 5: Confirming the Answer

By substituting back into our function form and checking with initial conditions and simplifying any arising identity, we confirm that it satisfies the original problem statement.

Key Concepts

Differentiable Function

In mathematics, a differentiable function is one that possesses a derivative at each point within its domain. A derivative essentially represents the function's rate of change and plays a critical role in calculus.
When solving functional equations like this one, it's important to know whether the function is differentiable because it assures a certain level of smoothness and predictability about the behavior of the function.
- Differentiability ensures continuity, meaning that small changes in the input lead to small changes in the output.- When a function is differentiable, we can use limits to study its behavior around specific points.
In the given problem, the hint is provided via the limit: \( \lim_{h \to 0} \frac{1}{h} f(h) = 3 \). This indicates that the function behaves linearly near zero, confirming its differentiability.
This piece of information allowed us to deduce that the derivative at zero, \( f'(0) \), is 3. Hence, defining the function to have linear tendencies close to zero.

Cauchy Equation

The Cauchy functional equation generally takes the form \( f(x+y) = f(x) + f(y) \). It is a fundamental equation in functional analysis and provides a way to understand the additivity property of functions while seeking solutions.
In this problem, the equation \(f(x+y) = f(x) + f(y) + xy\) includes an additional term, \(xy\). This modifies the typical Cauchy equation and indicates a mix of linear and quadratic components within the function \(f(x)\).
Understanding this is crucial because it directs how we manipulate and test potential forms for \(f(x)\), especially when mixed with differentiability conditions.
- The primary goal is to find a function form that satisfies both the equation in its entirety and the limit condition provided.- For solving, the function is expanded as \(f(x) = ax + bx^2\) based on the expectations from the equation's structure. This adjustment allows solving for values of \(a\) and \(b\), showing the relationship between linear and quadratic influence in forming the function.

Limit Evaluation

Evaluating limits is a crucial skill in calculus and analysis, key to understanding the behavior of functions as inputs approach particular values. In this problem, the condition \( \lim_{h \to 0} \frac{1}{h} f(h) = 3 \) offers substantial clues. It tells us about the function's slope at zero, essentially serving as derivative information.
- Limits help in identifying how functions behave infinitesimally close to a given point. This assists in approximating the function's immediate rate of change.- Here, evaluating the limit has shown that the derivative at zero is 3, indicating a strong linear component of the function. This insight guides the modification and verification of function forms, confirming that \(a = 3\) when dealing with a general form \(f(x) = ax + bx^2\).
The process of limit evaluation thus defines constraints on potential solutions, aligning function forms and derivatives to meet the problem's conditions.