Problem 132 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f\) be a real valued function satisfying \(f(x)+\) \(f(x+6)=f(x+3)+f(x+9) .\) Then, \(\int_{x}^{x+12} f(t) d t\) is (A) A linear function (B) An exponential function (C) A constant function (D) None of these

Step-by-Step Solution

Answer

The integral is a constant function (Option C).

Step 1: Understanding the Functional Equation

The given functional equation is \( f(x) + f(x+6) = f(x+3) + f(x+9) \). We need to explore this pattern to see how \(f\) behaves over intervals.

Step 2: Exploring the Behavior Over Intervals

Notice that the equation suggests some periodicity by repeating value differences over intervals. Testing values shows repetitive value substitutions indicating a specific behavior.

Step 3: Checking the Periodicity of the Function

By trying values and using the periodic property, it can be inferred that the function might behave periodically over a 12-unit interval. Substituting values in the original functional equation leads to reconfirming this periodic behavior.

Step 4: Evaluating the Integral

Given the periodic nature, compute \( \int_{x}^{x+12} f(t) \, dt \) over one period. The result of this integral is a constant due to the periodicity.

Key Concepts

Functional Equation

A functional equation is an equation in which the unknowns are functions rather than simple variables. To solve such equations, the goal is to find all functions that satisfy the given equation for all allowable input values. In our exercise, the functional equation is given as \( f(x) + f(x+6) = f(x+3) + f(x+9) \).
One crucial step in solving a functional equation is identifying any patterns or regularities in the expression. By manipulating the equation, we look for consistent behaviors of the function across different input values.
These insights, such as symmetry or periodicity, help in building an understanding of the functional form of the solution. Discovering such patterns is often key to unlocking the solution structure, and in some cases can lead to finding specific periodicities, as seen in our exercise.

Periodicity

Periodicity refers to the repetition of a function's values at regularly spaced intervals. A function \( f \) is said to be periodic if there exists a positive value \( T \) such that \( f(x) = f(x+T) \) for all \( x \). The smallest such \( T \) is the period of the function.

• In the context of our exercise, by inspecting the functional equation, \( f(x) + f(x+6) = f(x+3) + f(x+9) \), we can test and discover the behavior of the function over varying intervals.
• Through substitution and exploring potential periodic patterns, we surmise that the function is periodic with period 12.
• This conclusion is crucial for calculating integrals over intervals that span multiple periods, simplifying calculations.

Recognizing periodicity allows us to treat integrals over these intervals as sums of repeating segments, often simplifying the computation drastically.

Real Valued Function

A real valued function is a function that has real numbers as its range. This means that for every input (or \( x \)-value), the output (or \( f(x) \)) is a real number.
Real valued functions are essential in various fields of study and applications, dealing with any physical quantities that can be measured or quantified.

• In our given problem, the function \( f \) is defined to be real valued, which simplifies the analysis since we work entirely within the realm of real numbers.
• The real nature of the function ensures that properties like periodicity observed within the function can have tangible interpretations and are calculable.

Understanding the properties and definitions of real valued functions is fundamental, as it frames the problem in a context where we apply various mathematical techniques and ideas consistently.