Problem 117 - Calculus for JEE ADVANCED

Question

From the given data, $$ f(1-x)=f(1+x) $$ and \(f(2-x)=f(2+x)\) \(\ln (2)\), replacing \(x\) by \(1+x\), we have $$ f(1-x)=f(3+x) $$ or \(f(1+x)=f(3+x)\) or \(f(x)=f(2+x)\)

Step-by-Step Solution

Answer

The function is periodic with \( f(x) = f(2+x) \).

Step 1: Analyze the Function Property

We are given that \( f(1-x) = f(1+x) \). This means the function values are symmetric around 1, implying periodicity or reflection property in the function for these values.

Step 2: Apply the Second Given Condition

The equation \( f(2-x) = f(2+x) \) indicates that the function values are symmetric around 2, suggesting another periodicity or reflection around 2. This additional condition reinforces that \( f(x) \) has a symmetrical behavior.

Step 3: Substitute and Simplify

Substitute \( x = 1 + x' \) in the original function equation: \( f(1 - (1 + x')) = f(1 + (1 + x')) \). This simplifies to \( f(-x') = f(2 + x') \).

Step 4: Derive the Final Relation

Utilizing the expression \( f(-x') = f(2 + x') \) and combining the reflection properties, we identify that \( f(x) = f(2 + x) \). This result indicates the periodic nature of \( f \) with a shift symmetry by 2.

Key Concepts

Symmetry in Functions

Symmetry in functions refers to a consistent pattern where function values repeat in a predictable manner around a certain point. In the context of the given exercise, symmetry is evident from the fact that \( f(1-x) = f(1+x) \). This equation shows that for each value of \( x \), the function values at \( 1-x \) and \( 1+x \) are identical.

Understanding symmetry is crucial because it simplifies calculations and predictions. For instance, if a graph of the function is drawn, it would reflect itself around \( x = 1 \).

Symmetry can appear in various forms, like:

• Even Symmetry: Where \( f(x) = f(-x) \), leading to a reflection across the y-axis.
• Odd Symmetry: Where \( f(x) = -f(-x) \), resulting in rotational symmetry about the origin.
• More general symmetry: Such as the exercise's specific location symmetry, centered around some point different from zero or the origin.

Recognizing symmetry allows us to predict the function's behavior without needing to compute every individual value.

Periodicity

Periodicity in functions takes place when function values repeat at regular intervals. In this exercise, the condition \( f(x) = f(2+x) \) reflects periodicity with a shift length of 2.

To recognize periodicity, look for repeated patterns along the \( x \)-axis. Once identified, you can predict values and construct function graphs more efficiently.

Periodicity is mathematically expressed as:

• For some constant \( p \), a function is periodic if \( f(x) = f(x+p) \) for all \( x \).

This property is very useful because it implies that knowing the function over one interval can help understand its behavior over all other intervals, fitting into that period. Periodic functions often appear in real-world scenarios like sound waves and seasonal cycles, making them pivotal in modeling repetitive natural phenomena.

Reflection Property

The reflection property in functions is observed when outputs mirror themselves over a specific line or point. In this exercise, the reflection property is expressed through equations like \( f(1-x) = f(1+x) \) and \( f(2-x) = f(2+x) \). These show how the function duplicates its behavior symmetrically around specific points (in this case, 1 and 2).

Recognizing reflection in functions allows one to:

• Simplify function expressions and equations.
• Predict behavior due to established symmetry.
• Graphically understand how a function will look, as it helps indicate mirror lines.

The reflection property is akin to symmetry but is specific in highlighting how a function reflects its values across defined central points. This understanding aids in reducing computations and provides a visual and algebraic geometry insight, useful for problem-solving and analytical predictions.