Problem 64 - Essential Calculus Early Transcendentals

Question

Suppose that \(f^{\prime \prime \prime}\) is continuous and \(f^{\prime}(c)=f^{\prime \prime}(c)=0,\) but \(f^{\prime \prime \prime}(c)>0 .\) Does \(f\) have a local maximum or minimum at \(c ?\) Does \(f\) have a point of inflection at \(c ?\)

Step-by-Step Solution

Answer

There is no local maximum or minimum at \( c \), but \( f \) has a point of inflection there.

Step 1: Analyze the given conditions

We are given that \( f'(c) = 0 \) and \( f''(c) = 0 \). This implies that \( c \) is a critical point and also a potential inflection point. Furthermore, \( f'''(c) > 0 \) indicates the change in concavity immediately around \( c \).

Step 2: Determine the nature of the critical point using the third derivative test

Since \( f'(c) = 0 \) and \( f''(c) = 0 \), we apply the third derivative test to determine the nature of \( f \) at \( c \). The third derivative, \( f'''(c) > 0 \), indicates a change in concavity, suggesting that \( f \) transitions from concave down to concave up at \( c \).

Step 3: Conclusion about local extremum

Since the third derivative is positive and changes the concavity from down to up, there is no local extremum (neither maximum nor minimum) at \( c \). The function \( f \) does not achieve a maximum or minimum at this critical point.

Step 4: Conclusion about point of inflection

A point of inflection occurs when the concavity changes, which is confirmed by \( f'''(c) > 0 \). Thus, \( f \) has a point of inflection at \( c \) because the sign change in the concavity is indicated by the positive value of the third derivative.

Key Concepts

Point of Inflection

A point of inflection is where a function changes its concavity. It means the graph of the function changes from concave up to concave down or vice versa. This change indicates a shift in the way the curve bends. To identify a point of inflection, we often need to consider higher derivatives of the function.

In the given problem, the key indicator is the third derivative of the function, represented as \( f'''(c) \). For the point \( c \), we know:

• \( f'(c) = 0 \)
• \( f''(c) = 0 \)
• \( f'''(c) > 0 \)

The fact that \( f'''(c) > 0 \) suggests that the concavity is changing. Specifically, it denotes that the function transitions from concave down to concave up. As a result, \( c \) is indeed a point of inflection. This shift in the direction of curvature confirms the presence of a point of inflection at \( c \). Remember, at this point, while the function shows a change in the bending direction, it does not necessarily imply a maximum or minimum value of the function.

Critical Point

A critical point is where the derivative of a function is zero or undefined. It hints at possible maxima, minima, or points of inflection for the function. Critical points are essential in analyzing the behavior of functions and understanding where significant changes occur.

In our scenario, we learn that at \( c \):

• \( f'(c) = 0 \)
• \( f''(c) = 0 \)

These conditions satisfy both the first and second derivatives equating to zero. This qualifies \( c \) as a critical point. However, the situation is extended further by \( f'''(c) > 0 \), providing more insights. Given this context, the third derivative test helps affirm that, while \( c \) is a critical point, it does not indicate a local maximum or minimum, but rather a point of inflection. The presence of such additional conditions helps clarify the exact nature of the behavior of the function at this critical point.

Concavity Change

Concavity relates to the direction in which a function bends. A function can be concave up (like a cup) or concave down (like a cap). Understanding concavity helps in recognizing how steep or flat a function becomes over an interval.

The third derivative test is crucial in identifying changes in concavity at points where both the first and second derivatives equal zero. Here,

• \( f'(c) = 0 \)
• \( f''(c) = 0 \)
• \( f'''(c) > 0 \)

Here, the positive value of \( f'''(c) \) indicates that the concavity changes from concave down to concave up at point \( c \). This alteration confirms the point is a point of inflection, but not a local extremum. Such a change occurs without attaining a local maximum or minimum, showing the power of the third derivative test in analyzing and understanding a function's behavior during a concavity change. These observations highlight how even subtle forms of analysis can provide profound insights into the function's narrative conveyed through its derivatives.