Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f\) have a derivative on an interval \(I\). Prove that between successive distinct zeros of \(f^{\prime}\) there can be at most one zero of \(f .\) Hint: Try a proof by contradiction and use Rolle's Theorem (Problem 22).

Step-by-Step Solution

Answer

At most one zero of \( f \) exists between successive distinct zeros of \( f' \).

Step 1: Understand the Problem

We need to prove that between successive distinct zeros of the derivative function, \( f' \), there can be at most one zero of the function \( f \). This involves using a contradiction approach and applying Rolle's Theorem.

Step 2: Set Up a Proof by Contradiction

Assume for contradiction that there are two zeros of \( f \), say \( c \) and \( d \), in the interval \( (a, b) \), where \( a < c < d < b \) and \( f'(a) = f'(b) = 0 \). Our goal is to find a contradiction under this assumption.

Step 3: Utilize Rolle's Theorem

Rolle's Theorem states that if a function \( g \) is continuous on \([p, q]\) and differentiable on \((p, q)\) with \( g(p) = g(q) \), then there exists at least one \( r \) in \((p, q)\) such that \( g'(r) = 0 \).

Step 4: Apply Rolle's Theorem to \( f \) Between \( c \) and \( d \)

Given our assumption, since \( f(c) = f(d) = 0 \) and \( f \) is differentiable at least on \((c, d)\), Rolle's Theorem ensures there's a point \( x \) in \( (c, d) \) where \( f'(x) = 0 \).

Step 5: Analyze the Results

Given our assumption of two zeros and application of Rolle's Theorem, we find that there must be at least one zero of \( f' \) in \( (c, d) \). This contradicts the given that \( c \) and \( d \) are zeros of \( f \) and there are only zeros of \( f' \) at \( a \) and after \( b \).

Step 6: Conclude Contradiction

The contradiction arises because if \( c \) and \( d \) were both zeros of \( f \), there would be an additional zero of \( f' \) within \((a, b)\), contrary to our original setup. Hence, there can be at most one zero of \( f \) between successive distinct zeros of \( f' \).

Key Concepts

Derivative

A derivative represents the rate of change of a function with respect to a variable. In simple terms, the derivative tells us how a function's output value changes as we change its input value incrementally. Given a function \( f(x) \), the derivative is usually denoted by \( f'(x) \) or \( \frac{df}{dx} \).Understanding derivatives is crucial in calculus because they help us analyze the behavior of functions. They enable us to find:

• Slopes of tangent lines
• Rates of change (like velocity)
• Local maxima and minima of functions
• Inflection points

For example, if \( f'(x) = 0 \), this indicates that the function has a horizontal tangent at that point, which could mean a local minimum, maximum, or an inflection point. When the task asks us to examine the zeros of \( f' \), we're focusing on points where this rate of change is zero, signifying potential critical points for the function \( f \).

Zero of a Function

A zero of a function is any value of \( x \) for which the function takes a value of zero, i.e., \( f(x) = 0 \). These points are important because they show where the function intersects the x-axis.Zeros of a function are critical in understanding its behavior and graph shape. They can indicate where the function changes sign, providing insight into the function’s increasing or decreasing nature.

• Zeros can represent roots or solutions to equations.
• They can signal critical points when dealing with derivative functions.
• They help in practically applying concepts like Rolle's Theorem, where such zeros define intervals of interest.

For a derivative function \( f'(x) \), zeros indicate points where the original function \( f(x) \) doesn't change its rate, suggesting pauses in increases or decreases, often at peaks or troughs.

Proof by Contradiction

Proof by contradiction is a logical method used to demonstrate the truth of a statement by assuming the opposite condition leads to a contradiction. Here's how it works:

• Start by assuming the opposite of what you want to prove.
• Through logical steps, derive a statement that is impossible or contradicts a known fact.
• Conclude that since the assumption leads to a contradiction, the original statement must be true.

In our exercise, to prove there can be at most one zero of \( f \) between successive zeros of \( f' \), we assume there are two such zeros. This leads us to a logically impossible situation, demonstrated using Rolle's Theorem. This contradiction helps us affirm that our initial claim — there can be only one such zero of \( f \) — holds true.

Calculus Theorems

Calculus theorems provide foundational insights into analyzing functions and determining their properties. Rolle's Theorem is one of these essential tools that assists in understanding behavior over an interval.Rolle's Theorem states:"If a function \( g(x) \) is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( g(a) = g(b) \), then there exists at least one \( c \) in \( (a, b) \) such that \( g'(c) = 0 \)."Key takeaways from calculus theorems include:

• They allow predictions about the function's critical points based on its boundary behavior.
• They extend into other powerful theorems like the Mean Value Theorem.
• They work as tools for proof strategies, like proof by contradiction, within calculus problems.

In the given problem, Rolle's Theorem helps identify potential zeros of the derivative, lending structure to the proof method and ensuring a rigorous approach to find contradictions in assumptions.