Problem 52 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f: R \rightarrow R\) be a function such that \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}, f(0)=0\) and \(f^{\prime}(0)=3\) Then, (A) \(f(x)\) is a quadratic function (B) \(f(x)\) is continuous but not differentiable (C) \(f(x)\) is differentiable in \(R\) (D) \(f(x)\) is bounded in \(R\)

Step-by-Step Solution

Answer

(C) \( f(x) \) is differentiable in \( R \).

Step 1: Analyze the Functional Equation

The given functional equation is \( f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3} \). This is a type of Cauchy functional equation for additive functions. We need to explore if this can lead us to determine whether \( f(x) \) is linear or not.

Step 2: Check the Initial Condition

We know that \( f(0) = 0 \). This is important since it provides a starting point for evaluating how the function behaves around zero.

Step 3: Use the Derivative Condition

We are given that \( f'(0) = 3 \). This indicates the rate of change of \( f(x) \) at zero, suggesting that locally around zero, \( f(x) \) resembles a linear function with a slope of 3.

Step 4: Consider Behavior of Linear Functions

If \( f(x) \) were linear, say \( f(x) = cx \), it would satisfy both \( f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3} \) and \( f'(0) = 3 \). Substituting gives \( c\left(\frac{x+y}{3}\right) = \frac{cx + cy}{3} \), which is true, so this fits the model of a linear function.

Step 5: Confirm Differentiability and Calculate the Function Form

Since \( f'(0) = 3 \), and assuming \( f(x) \) is linear, we conjecture \( f(x) = 3x \). This function is differentiable as it is linear, and its derivative is constant, satisfying \( f'(x) = 3 \) everywhere.

Key Concepts

Differentiability in Functions

The concept of differentiability is crucial in understanding how functions change. A function is differentiable at a point if it has a derivative there, which means it's smooth and has no sharp corners or cusps. In our exercise, the function is given to be differentiable at zero with \(f'(0) = 3\). This tells us that the slope, or the rate of change of the function, at \(x = 0\) is 3. This matches what we'd expect from a linear function, which has the same slope everywhere.

Differentiability is not just about having derivatives at all points, but it implies continuity as well. A differentiable function on an interval is always continuous on that interval. Here, differentiability points to the possibility that the function is consistently changing without interruption or irregular jumps. This smooth behavior is a key indication that the function could be simpler, potentially linear.

• A function like \(f(x) = 3x\), being differentiable everywhere, exemplifies smooth and uniform change.
• The derivative of a linear function is constant, showing consistent slope throughout its domain.

Understanding differentiability helps us predict and analyze the behavior of functions in various situations, especially in calculus and real analysis.

Understanding Linear Functions

Linear functions are among the simplest types of functions and form a straight line when graphed. They can be expressed in the form \(f(x) = cx\), where \(c\) is the constant rate of change or slope of the line. In our problem, we're exploring if the function \(f(x)\) is linear under the given conditions.

Step 4 of the solution demonstrates the testing of the linearity of \(f(x)\). Substituting \(f(x) = cx\) into the given functional equation, we see that it satisfies \(f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3}\). This condition holds true for linear functions, as they add and scale in a straightforward, predictable manner.

• The simplicity of linear functions is due to their uniform rate of change, making calculations and predictions easy.
• In calculus, linear functions are often used to approximate more complex functions locally due to their easy properties.

Finding that a function is linear simplifies many analyses, as we know its derivative is constant, and it's easy to manipulate and predict its behavior, given any input.

Analysis of Functions

Analyzing functions involves exploring their properties like continuity, differentiability, and bounds. Let's break down some key analyses relevant to our exercise:

Continuity:
The concept of continuity is tied to differentiability; a differentiable function on a domain is also continuous on that domain. This means there are no sudden jumps or breaks in the function. For \(f(x) = 3x\), which we suspect for our function, this would be true as it's defined for all real numbers smoothly.

Bounds:
Being linear, \(f(x) = 3x\) is not bounded in \(\mathbb{R}\) — it gets both infinitely large and small as \(x\) does. This is a typical feature for linear functions, except in specific domains where we force bounds.

• Analysis boils down to understanding the full behavior of a function over the domain we care about.
• Boundedness and continuity have implications on function integration and limits.

Comprehensive analysis helps us determine how a function behaves in all scenarios, giving a clearer picture of what to expect and preparing us to apply these functions in practical problems.