Problem 36 - Essential Calculus Early Transcendentals

Question

A number \(a\) is called a fixed point of a function \(f\) if \(f(a)=a .\) Prove that if \(f^{\prime}(x) \neq 1\) for all real numbers \(x\) then \(f\) has at most one fixed point.

Step-by-Step Solution

Answer

Since \( f'(x) \neq 1 \), \( f \) can have at most one fixed point.

Step 1: Understanding the Problem

We need to prove that a function \( f \) has at most one fixed point, given that its derivative \( f'(x) eq 1 \) for all real numbers \( x \). A fixed point \( a \) satisfies \( f(a) = a \).

Step 2: Assume Two Fixed Points

Assume for contradiction that \( f \) has two distinct fixed points, \( a \) and \( b \), such that \( f(a) = a \) and \( f(b) = b \).

Step 3: Apply the Mean Value Theorem

By the Mean Value Theorem, since \( f \) is differentiable, there exists some \( c \) between \( a \) and \( b \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \).

Step 4: Calculate the Slope

Given \( f(a) = a \) and \( f(b) = b \), the expression becomes \( f'(c) = \frac{b - a}{b - a} = 1 \).

Step 5: Contradiction Arises

Since \( f'(c) eq 1 \), there is a contradiction. Thus, \( f \) cannot have two distinct fixed points.

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus which bridges the gap between derivatives and the behavior of functions over an interval. It specifies that for a function \( f \) that is continuous over a closed interval \([a, b]\) and differentiable over the open interval \((a, b)\), there exists at least one point \( c \) in \((a, b)\) where the slope of the tangent (derivative) at \( c \) is equal to the slope of the secant line joining the endpoints. In mathematical terms:

• \( f'(c) = \frac{f(b) - f(a)}{b - a} \)

This theorem helps in understanding the behavior of functions and provides essential insights into their properties. In our exercise, the application of MVT led to the realization of a contradiction when assuming multiple fixed points, as it resulted in the impossible condition \( f'(c) = 1 \), while our condition was \( f'(x) eq 1 \) everywhere.

Differentiable Functions

Differentiable functions are functions that have a derivative at every point in their domain. This means the function has a unique tangent at every point, and it implies the function is smooth without any sharp corners or edges.

• If a function is differentiable on an interval, it is also continuous on that interval.
• Not all continuous functions are differentiable (consider a function with a sharp corner).

In the context of our problem, differentiability ensures that the Mean Value Theorem can be applied. This prerequisite is crucial because MVT relies on the concept of a derivative, which in turn depends on differentiability. When analyzing fixed points under the constraint \( f'(x) eq 1 \), differentiability plays a key role in making the argument about at most one fixed point valid.

Derivative

The derivative of a function represents the rate of change of the function's output with respect to changes in its input. In simpler terms, it's the slope of the function at any given point. The notation \( f'(x) \) is often used to denote the derivative of \( f \) at \( x \).

• A positive derivative indicates that the function is increasing.
• A negative derivative suggests the function is decreasing.
• If the derivative is zero, the function has a horizontal tangent (a potential maximum or minimum).

For discussing fixed points, understanding the derivative helps in examining the local behavior of the function. In our situation, knowing that \( f'(x) eq 1 \) for any \( x \) is critical because it directly impacts the proof that the function cannot have two distinct fixed points without reaching a contradiction.

Unique Fixed Point

A unique fixed point is a value \( a \) for which \( f(a) = a \), and importantly, it is the only such value for the function within its domain. When a function satisfies the condition that its derivative is not equal to 1 everywhere (\( f'(x) eq 1 \)), it means any fixed point it has is unique.

• If two distinct fixed points existed, applying the Mean Value Theorem would imply the derivative equals 1, which contradicts our condition.
• Thus, ensuring \( f'(x) eq 1 \) secures the function against any more than one fixed point.

In real-world applications, identifying fixed points and ensuring their uniqueness can be crucial for understanding stability and equilibrium in systems, such as in economics or population dynamics.