Problem 61 - Essential Calculus Early Transcendentals

Question

Prove that if \((c, f(c))\) is a point of inflection of the graph of \(f\) and \(f^{\prime \prime}\) exists in an open interval that contains \(c\) , then \(f^{\prime \prime}(c)=0 .\) [Hint. Apply the First Derivative Test and Fermat's Theorem to the function \(g=f^{\prime} . ]\)

Step-by-Step Solution

Answer

\(f^{\prime\prime}(c) = 0\) at the point of inflection.

Step 1: Understand a Point of Inflection

A point of inflection \(c, f(c)\) is where the concavity of the graph changes. This means that the second derivative \f^{\prime\prime}\ changes sign at \c\.

Step 2: Define the Function g(x)

Let \g(x) = f^{\prime}(x)\. This means \g(x)\ is the first derivative of \f(x)\. A point of inflection on \(f\) occurs when \g(x)\ changes from increasing to decreasing or vice versa.

Step 3: Apply the First Derivative Test to g(x)

The First Derivative Test states that if the sign of the derivative changes, there is a local extremum. Since \(c\) is a point of inflection, \g^{\prime}(c)=f^{\prime\prime}(c)\ changes signs, indicating either a maximum or minimum at \(c\).

Step 4: Apply Fermat's Theorem

Fermat's Theorem states that if \g(x)\ has a local extremum at \(c\) and \g^{\prime}(c)\ exists, then \g^{\prime}(c) = 0\. Since \(c\) is a point of inflection and \g(x)\ has an extremum at \(c\), \f^{\prime\prime}(c) = 0\.

Step 5: Conclude the Proof

Since \f^{\prime\prime}(c) = 0\ by Fermat's Theorem and the First Derivative Test, it's proved that the second derivative must be zero at the point of inflection when it exists in an open interval containing \(c\).

Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), offers powerful insights into the behavior of the original function \( f(x) \). It primarily tells us about the concavity of the graph.

• If \( f''(x) > 0 \), the graph of \( f(x) \) is concave up, resembling the shape of a cup.
• If \( f''(x) < 0 \), the graph is concave down, similar to an upside-down cup.
• If \( f''(x) = 0 \), we may have a point of inflection, but further analysis is needed to confirm it.

At a point of inflection, the concavity of the graph changes direction. For the second derivative to actually indicate a point of inflection at \( c \), not only should \( f''(c) = 0 \), but \( f''(x) \) should change sign around \( c \). This change indicates that the function goes from concave up to concave down, or vice versa. Understanding this sign change is crucial for identifying genuine points of inflection.

First Derivative Test

The First Derivative Test is a valuable tool used to identify local extrema, like peaks (maxima) and valleys (minima), of a function. It revolves around how the first derivative \( f'(x) \) behaves.

• When \( f'(x) \) changes from positive to negative, the function \( f(x) \) has a local maximum.
• When \( f'(x) \) changes from negative to positive, \( f(x) \) has a local minimum.
• If no sign change occurs, then there's no local extremum at that point.

In the context of the exercise, we consider the function \( g(x) = f'(x) \), which is the derivative of \( f(x) \). By applying the First Derivative Test to \( g(x) \), we observe the behavior of \( g'(x) = f''(x) \). If \( f''(x) \) changes signs at \( c \), it implies a local extremum of \( g(x) \), thereby suggesting \( c \) is a point of inflection for \( f(x) \).

Fermat's Theorem

Fermat's Theorem plays a key role in finding the extrema of a function. It states that if a function \( g(x) \) reaches a local extremum at a point \( c \) and the first derivative exists at \( c \), then \( g'(c) = 0 \).

Applying this to our exercise, let's take \( g(x) = f'(x) \). If \( g(x) \) has a local extremum at \( c \), then according to Fermat's Theorem, the derivative of \( g(x) \), which is \( g'(c) = f''(c) \), must equal zero.

This result is crucial because it aligns with our understanding of a point of inflection—demonstrating that for \( c \) to be a point of inflection of \( f(x) \), not only must \( f''(x) \) change signs, but \( f''(c) \) must also be zero. Fermat's Theorem confirms this by ensuring that \( g'(c) = 0 \), making it clear that \( c \) is indeed a point of inflection.