Problem 67 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(C) \(f(x) = 3x + \frac{x^2}{2}\).

Step 1: Substitute Known Expression in Functional Equation

Given the functional equation \( f(x+y) = f(x) + f(y) + xy \). Substitute \( y = 0 \):\[ f(x+0) = f(x) + f(0) + x \cdot 0 \Rightarrow f(x) = f(x) + f(0) \].Thus, \( f(0) = 0 \).

Step 2: Evaluate Derivative Using Limit

The condition \( \lim_{h \to 0} \frac{1}{h} f(h) = 3 \) suggests we find \( f'(0) \). We use the derivative definition:\[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h)}{h} = 3 \].

Step 3: Differentiate the Functional Equation

Differentiate the given functional equation with respect to \( y \):\[ \frac{d}{dy} (f(x+y)) = \frac{d}{dy} (f(x) + f(y) + xy) \]This gives \( f'(x+y) = f'(y) + x \).

Step 4: Simplify Using Known Values

Setting \( y = 0 \), we get \( f'(x+0) = f'(0) + x \). Therefore,\[ f'(x) = 3 + x \].

Step 5: Integrate to Find Original Function

Integrate \( f'(x) = 3 + x \) with respect to \( x \):\[ f(x) = \int (3 + x) \, dx = 3x + \frac{x^2}{2} + C \].

Step 6: Determine Constant Using Initial Condition

Since \( f(0) = 0 \), substitute into \( f(x) = 3x + \frac{x^2}{2} + C \):\[ f(0) = 3 \cdot 0 + \frac{0^2}{2} + C = 0 \Rightarrow C = 0 \].

Step 7: Conclusion and Answer

The function \( f(x) = 3x + \frac{x^2}{2} \) satisfies all conditions. Comparing with the options given, the correct answer is option (C).

Key Concepts

Functional Equation

A functional equation is very much like an algebraic equation, but instead of involving numbers, it involves functions. It describes the relation between functions or between the values of the functions for given values of variables. A clever use of functional equations can reveal properties of unknown functions.
In this exercise, we're given a functional equation: \[f(x+y) = f(x) + f(y) + xy\]This tells us how the function behaves when two arguments are added. To explore this, substitute simple values, such as setting one of the variables to zero. By substituting \(y = 0\), we found that \(f(0) = 0\). This is a typical approach for getting insights into the function's behavior.

Limit of a Function

Limits are a fundamental concept in calculus and analysis, describing the behavior of functions as they approach a certain point. The limit \[\lim_{h \rightarrow 0} \frac{1}{h} f(h) = 3\]gives important information about the function's derivative at zero.Understanding limits helps in establishing continuity and differentiability of functions. Here, the limit implies that as \(h\) approaches zero, the ratio approaches 3, which indicates the slope of the tangent line at zero or the value of the derivative at that point.

Derivative

The derivative of a function measures how the function's output changes as its input changes. In calculus, it's the mathematical representation of the rate of change or slope of the function.From our functional equation, differentiating with respect to \(y\), and utilizing the limit, we deduced\[f'(0) = 3\]and then more generally,\[f'(x) = 3 + x\]This tells us about the rate of change for the function across its entire domain. Derivatives are essential for understanding the turning points and concave structure of graphs.

Integration

Integration is often seen as the reverse process of differentiation. While derivatives give us information about the rate of change, integration helps us to find the original function given its derivative.In this problem, to find the function \(f(x)\), we integrated its derivative:\[\int (3 + x) \, dx = 3x + \frac{x^2}{2} + C\]Through integration, we reconstructed the original formula of the function by determining the constant \(C\) using initial conditions. This process encapsulates the power of integration, capturing the cumulative effect of a variable's change over an interval.