Problem 114 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x+p)=1+\left[2-3 f(x)+3(f(x))^{2}-(f(x))^{3}\right]^{1 / 3}\), \(\forall x \in R\), where \(p>0\). Then, \(f(x)\) is periodic with period. (A) \(p\) (B) \(2 p\) (C) \(4 p\) (D) None of these

Step-by-Step Solution

Answer

The period of the function is \( 2p \).

Step 1: Analyze the Functional Equation

Given the function equation \( f(x + p) = 1 + \left[ 2 - 3f(x) + 3(f(x))^2 - (f(x))^3 \right]^{1/3} \). We need to find such \( f(x) \) which satisfies this equation. Let's assume periodicity with period \(p\) or multiples and check possibilities.

Step 2: Simplify the Right-Hand Expression

Set \( y = f(x) \) to simplify the expression: \( f(x + p) = 1 + \left[ 2 - 3y + 3y^2 - y^3 \right]^{1/3} \). The expression inside the cube root is actually \( (1-y)^3 \), after expansion: \( 1 - 3y + 3y^2 - y^3 \). So, \( f(x+p) = 1 + (1-y) = 2 - y \).

Step 3: Test for Period \( p \)

If \( f(x) = y \), then \( f(x + p) = 2 - y \). For periodicity with period \( p \), \( f(x) \) should be equal after one cycle: \( f(x) = f(x + p) \). This implies \( y = 2 - y \), leading to \( y = 1 \). Thus, constant function \( f(x) = 1 \) is a solution, suggesting periodicity of \( p \).

Step 4: Generalize for Other Periods

Checking for periodicity with \( 2p \), iterate the functional equation: \( f(x + 2p) = f((x + p) + p) = 2 - f(x + p) \), which simplifies to \( f(x + 2p) = y \). Thus \( f(x) \) returns to the original \( y \) value after \( 2p \), indicating periodicity of \( 2p \).

Step 5: Conclude the Periodicity

As \( f(x) \) behaves such that \( f(x + 2p) = f(x) \), and doesn't conform to period \( p \) alone (as \( f(x) = 1 \) isn't variable-dependent), the smallest periodicity is \( 2p \). Therefore, \( f(x) \) is periodic with period \( 2p \).

Key Concepts

Functional Equations

A functional equation is an equation where the variables can include not only numbers but also functions. In these equations, you must find a function or functions that satisfy the given relationships. In this specific exercise, the functional equation given is a transformation involving the function itself:

• It translates a function to another point, creating a new expressional form.
• The equation: \( f(x+p) = 1 + \left[ 2 - 3f(x) + 3(f(x))^2 - (f(x))^3 \right]^{1/3} \) showcases the relationship of \( f(x+p) \) with \( f(x) \).
• Solving requires setting up equations, simplifying expressions, or using assumptions like potential periodicity (e.g., \( p \), \( 2p \), etc.).

It's essentially a puzzle where you solve by determining the function's form. By testing these assumptions with initial values (like assuming \(f(x)\) could be constant), you explore possible solutions.

Periodicity

Periodicity in mathematics refers to the repeating nature of a function at regular intervals. A function \( f(x) \) is periodic if there is a positive number \( p \), where \( p \) is called the period, such that \( f(x + p) = f(x) \) for all \( x \).

• The goal of finding periodicity involves exploring values of \( p \) that satisfy the relation of repetition.
• In the exercise, periodicity checks were crucial to confirm that \( f(x) = f(x + 2p) \), ensuring repetition after \( 2p \).
• By determining \( f(x+p) = 2-y \), and then iterating again for \( f(x + 2p) \), one confirms the simplest case of periodicity which fits the functional equation.

Understanding periodicity helps identify simplified function forms that conform to specific cycles, ensuring the function remains predictable and manageable.

Mathematical Analysis

Mathematical analysis involves the rigorous investigation and dissection of mathematical problems. In this context, mathematical analysis helps:

• Breakdown complex functions into simpler components that reveal underlying patterns or behaviors.
• Assess the behavior of a function to determine its periodic nature, making it easier to prove necessary properties of mathematical relationships.
• The analysis here showed that by setting \( y = f(x) \), the equation simplifies, exposing that internally, the cycle repeats with specific transformations, indicating periodic nature (i.e., \( f(x + 2p) = f(x) \)).

Careful mathematical analysis thus allows us to see beyond the surface complexity and to derive meaningful conclusions about the function properties.